This problem has two aspects: fiqhi and social/psychological. 

With regard to the fiqhi aspect, there are a number of issues: 

1.

The ruling on getting married without a wali (guardian), and the ruling on the child who is born as a result.

The marriage of a woman without a wali is an invalid marriage according to the majority of scholars, apart from Abu Haneefah (may Allah have mercy on him). 

The evidence quoted by the majority to support this view is as follows: 

The hadeeth of ‘Aa’ishah (may Allah be pleased with her) who said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Any woman who gets married without the permission of her guardians, her marriage is invalid”– he said this three times – “but if he consummates the marriage with her, then she is entitled to the mahr (dowry) because of the intimacy he had with her. Then if they differ, the ruler is the guardian of the one who has no guardian.”

Narrated by Abu Dawood (2083) and at-Tirmidhi (1102). Classed as saheeh by al-Albaani in Irwa’ al-Ghaleel, 6/243 

The hadeeth of Abu Moosa al-Ash‘ari (may Allah be pleased with him), according to which the Prophet (blessings and peace of Allah be upon him) said: “There is no marriage except with a guardian.”

Narrated by Abu Dawood (2085) and at-Tirmidhi (1101). Classed as saheeh by al-Albaani in Irwa’ al-Ghaleel, 6/235 

But the child who is born as a result of this invalid marriage is to be attributed to his father because it is an ambiguous marriage. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him) said: 

The marriage contract done without the presence of a guardian is an invalid marriage contract according to the correct opinion, which is that of the majority of scholars. The woman does not have the right to do the marriage contract on her own behalf. The Prophet (blessings and peace of Allah be upon him) said: “There is no marriage except with a guardian.” And he (blessings and peace of Allah be upon him) said: “No woman can give herself in marriage, and no woman can give another woman in marriage.” Both are saheeh hadeeths. But if they do that and the woman becomes pregnant, then the child is to be attributed to his father, because the contract is ambiguous. 

Noor ‘ala ad-Darb by Ibn Baaz, 20/197-198 

2.

The ruling on marriage of a man to a woman who is pregnant as a result of an improper marriage to someone else. 

Two issues are involved here: 

(i)If a woman has got married with an improper marriage contract, it is not permissible for her to marry anyone else until the first husband divorces her or the marriage contract is annulled.

Ibn Qudaamah (may Allah have mercy on him) said: If a woman gets married in an improper marriage contract, it is not permissible for her to get married to someone else until (the first one) divorces her or her marriage is annulled. 

Al-Mughni, 9/351 

(ii)It is not permissible for a man to marry a woman who is pregnant from another man until she gives birth, and if he does marry her, then it is an invalid marriage.

It says in Fataawa al-Lajnah ad-Daa’imah (18/246): 

In the case of a pregnant woman who is divorced or recently widowed, her ‘iddah lasts until she gives birth, because Allah, may He be exalted, says (interpretation of the meaning): “And for those who are pregnant (whether they are divorced or their husbands are dead), their ‘Iddah (prescribed period) is until they deliver (their burdens)” [at-Talaaq 65:4]. Any marriage contract done with such a woman is invalid and does not count as a marriage. End quote. 

From the above, it is clear that your friend has committed a number of actions that are contrary to sharee‘ah. She has to repent to Allah, may He be exalted, and put right her mistakes by telling her family about her previous marriage and the pregnancy that resulted from it, and she should separate from her second husband until she gives birth. Then after that, if she wants to stay with the first husband from whom she became pregnant, then she has to convince her father, and then do a (new) marriage contract, whether he had divorced her or not, because the first marriage contract is in fact invalid. But if she wants to stay with the second husband, then if the first husband had not divorced her, he must issue a divorce or annul the marriage, then she can do the marriage contract with the second husband. 

(iii)The ruling on forcing a girl to marry someone she does not want as her husband.

It is not permissible for the father to force his adult daughter to marry someone she does not want as a husband. 

That is because of the hadeeth of Abu Hurayrah (may Allah be pleased with him) according to which the Prophet (blessings and peace of Allah be upon him) said: “A previously married woman should not be given in marriage until she is consulted, and a virgin should not be given in marriage until her permission has been sought.” They said: O Messenger of Allah, how does she give permission? He said: “If she remains silent.”

Narrated by al-Bukhaari, 5136 

It was narrated from Ibn ‘Abbaas (may Allah be pleased with him) that a young woman came to the Prophet (blessings and peace of Allah be upon him) and told him that her father had given her in marriage against her wishes. The Prophet (blessings and peace of Allah be upon him) gave her the choice (between staying in the marriage or having it annulled).

Narrated by Abu Dawood, 2096. Classed as saheeh by al-Albaani in Saheeh Abi Dawood, 1/586 

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: 

With regard to the father forcing his adult virgin daughter into a marriage, there are two well-known scholarly opinions, both of which were narrated from Ahmad. 

The first says that he may force his adult virgin daughter, as is the view of Maalik and ash-Shaafa‘i. This is also the view favoured by al-Khuraqi, al-Qaadi and his companions. 

The second view is that he may not force her, as is the view of Abu Haneefah and others. This is also the view favoured by Abu Bakr ‘Abd al-‘Azeez ibn Ja‘far. This view is the correct one … Because it is proven in as-Saheeh that the Prophet (blessings and peace of Allah be upon him) said: “No virgin should be given in marriage until her permission is sought, and no previously married woman (should be given in marriage) until she is consulted.” It was said to him: What if the virgin feels too shy (to say anything)? He said: “Her permission is her silence.” According to another version narrated in as-Saheeh: “The virgin’s permission should be sought by her father.” This is a prohibition of the Prophet (blessings and peace of Allah be upon him): she should not be given in marriage until her permission is sought. This applies to the father and others. That is clearly stated in the other saheeh report, and the father himself should seek her permission. 

Moreover, the father does not have the right to dispose of her wealth without her permission if she is an adult of sound mind, and the issue of intimacy is more serious than her wealth, so how can it be permissible for him to give her in marriage against her wishes when she is mature and of sound mind? 

With regard to giving her in marriage even though she objects to the marriage, this is contrary to Islamic teaching and common sense, and Allah does not allow her guardian to force her to buy or sell or rent (her own property) except with her permission, or to eat or drink or wear anything she does not want, so how can he force her into being intimate with and living with someone she does not want to be intimate with or live with? Allah has ordained love and compassion between husband and wife, so if the marriage came about despite the fact that she did not want it at all, where is the love and compassion in that?

End quote from Majmoo‘ al-Fataawa, 23/22-25 

Ibn al-Qayyim (may Allah have mercy on him) said: 

It is proven in as-Saheeh that [the Prophet (blessings and peace of Allah be upon him)] said: “A virgin should not be given in marriage until her permission has been sought.” They said: O Messenger of Allah, how can she give permission? He said: “If she remains silent.” In Saheeh Muslim it says: “The virgin should be asked permission concerning her (marriage), and her permission is her silence.” 

This ruling dictates that an adult virgin should not be forced to marry and she should not be given in marriage except with her consent. This is the view of the majority of the early generations, and it is the view of Abu Haneefah and Ahmad, according to one of the reports from him. This is the view which we believe is correct, and we do not believe in anything other than that. It is in accordance with the ruling of the Messenger of Allah (blessings and peace of Allah be upon him) and with his commands and prohibitions, the basic principles of his sharee‘ah and the best interests of his ummah.

End quote from Zaad al-Ma‘aad, 5/88 

Shaykh Ibn Baaz (may Allah have mercy on him) was asked: Is it permissible for a father to force his daughter to marry someone she does not want? 

The answer was: 

Neither the father nor anyone else has the right to force his female relative who is under his guardianship to marry someone she does not want. Rather it is essential to seek her permission, because the Messenger (blessings and peace of Allah be upon him) said: “No previously married woman is to be given in marriage until she is consulted, and no virgin is to be given in marriage until her permission is sought.” They said: O Messenger of Allah, what is her permission? He said: “If she remains silent.” So what the father must do is seek her permission if she has reached the age of nine or more. By the same token, her guardians should not give her in marriage except with her permission. This is what is required of all, and if anyone has got married without permission, then the marriage is not valid, because one of the conditions of marriage is the consent of both spouses. If he has given her in marriage without her consent and has forced her with stern threats or beatings, then the marriage is not valid. If the husband knows that she does not want him he should not go ahead with it, even if the father does not care about that. What he must do is fear Allah, and not go ahead with marriage to a woman who does not want him, even if her father claims that he did not force her. What he must do is be cautious not to do that which Allah has forbidden to him, because the Messenger (blessings and peace of Allah be upon him) has instructed that her permission must be sought. We also advise the woman to fear Allah and to agree if her father wants to arrange her marriage, if the suitor is good in terms of his religious commitment and character. 

End quote from Majmoo‘ Fataawa wa Maqaalaat Mutanawwi‘ah, 20/414-415 

So you have to sincerely advise this father and inform him of this shar‘i ruling, and tell him that what is required of the Muslim is to follow the guidance of the Prophet (blessings and peace of Allah be upon him), and to discard any societal customs that are contrary to that. He should look at what is in his daughter’s best interests and look for someone whose religious commitment and character are good, and whom his daughter will accept as a husband. He is responsible for her because she is part of his flock, and the Prophet (blessings and peace of Allah be upon him) said: “Each of you is a shepherd and each of you is responsible. The ruler is a shepherd and is responsible. A man is the shepherd of his household and is responsible. A woman is the shepherd of her husband’s house and is responsible. The slave is the shepherd of his master’s wealth and is responsible. Each of you is a shepherd and each of you is responsible.”

Narrated by al-Bukhaari, 5188, from the hadeeth of Ibn ‘Umar (may Allah be pleased with him). 

To sum up the above: 

Her second marriage is definitely invalid, and there is no way it can be valid, and she should not let him be intimate with her; rather she must inform him of the facts of the matter. 

As for her first marriage, it is invalid according to the majority of scholars, and they have to repeat the marriage contract in the correct manner, if they want to remain married, or else annul the marriage and separate, if they want to end it. (In that case), she has to observe ‘iddah following the end of the marriage, then she may marry whomever she wants after that. See the answer to question no. 142177

With regard to the social/psychological aspect: 

(a)

you should advise your friend that the best way to solve family problems is to fear Allah, may He be exalted, as Allah has stated and confirmed in Soorat at-Talaaq, which is a soorah that deals with family problems, repeatedly enjoining people to fear Allah and stating that this is the way out and the key to provision. Allah, may He be exalted, says (interpretation of the meaning):

“And whosoever fears Allah and keeps his duty to Him, He will make a way for him to get out (from every difficulty).

3. And He will provide him from (sources) he never could imagine. And whosoever puts his trust in Allah, then He will suffice him. Verily, Allah will accomplish his purpose. Indeed Allah has set a measure for all things”

[al-Talaaq 65:2-3]. 

Part of fearing Allah, may He be exalted, is striving to please one’s parents; not doing anything for which Islam does not give permission; and repenting to Allah, may He be exalted. 

(b)

Family problems, such as that which has happened to your friend – may Allah guide her to that which is in her best interests in this world and the Hereafter – may be solved by being honest and truthful, because the effects of her action are not limited to her alone; rather it also affects the people around her and affects issues of lineage and rights. Maybe relatives or friends of the father’s whom he respects can intervene in order to tell him what has really happened and bring about reconciliation between them. 

(c)

Your friend should not build her hopes on imaginary love; no matter how strong the love between the genders may be, in the end it is merely an emotion, and emotions change quickly if they are based on whims and desires or worldly matters. The love that is likely to continue, on which a solid family may be built, is that which pays attention to the laws of Allah, may He be exalted, and which a person proceeds with after consulting others and praying to Allah for guidance (istikhaarah). 

And Allah knows best.