Firstly: 

The time for praying Witr ends when dawn comes, because the Prophet (peace and blessings of Allaah be upon him) said: ‘The night prayers are two by two, then when you fear that dawn is about to break, then pray one (rak’ah) and make the prayers that you have offered odd-numbered.” Narrated by al-Bukhaari, 472. 

Muslim (754) narrated from Abu Sa’eed (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said: “Pray Witr before dawn comes.”

If the adhaan for Fajr is given and a person has not yet prayed witr, he should delay it until the forenoon, after the sun has risen high, then he should pray whatever he can, two or four rak’ahs or more, two by two. If his habit is to pray three and he did not pray them at night, he should pray them in the forenoon with four rak’ahs and two tasleems. If his habit is to pray five and he was not able to do them at night because he was sick or he was asleep and so on, he should pray them in the forenoon with six rak’ahs and three tasleems, and so on, because the Prophet (peace and blessings of Allaah be upon him) used to do that. He used to pray witr with eleven rak’ahs, but if sickness or sleep kept him from doing that, he would pray them during the day with twelve rak’ahs. This is what ‘Aa’ishah (may Allaah be pleased with her) said according to the report narrated from her by the two shaykhs al-Bukhaari and Muslim, and this is what is prescribed for the ummah, following the example of the Prophet (peace and blessings of Allaah be upon him). 

Majmoo’ Fataawa Ibn Baaz, 11/300 

Shaykh Ibn Baaz was also asked: Should the last of Witr prayer be done when the adhaan begins for Fajr or at the end of the adhaan? If a person sleeps and misses it, should he make it up and how? 

He replied: 

It is prescribed for every believer, male and female, to pray Witr every night. The time for it is between ‘Isha’ prayer until dawn begins, because it is narrated in al-Saheehayn from Ibn ‘Umar that the Prophet (peace and blessings of Allaah be upon him) said: “The night prayers are two by two, then if one of you fears that dawn is about to break, let him pray one rak’ah to make what he has prayed odd-numbered.” Muslim narrated in his Saheeh that Abu Sa’eed al-Khudri said: “Pray witr before dawn comes.” Imam Ahmad, Abu Dawood and al-Tirmidhi narrated, in a report that was classed as saheeh by al-Haakim, from Khaarijah ibn Hudhaafah (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said: “Allaah has blessed you with a prayer that is better for you than red camels.” We said: O Messenger of Allaah, what is it? He said: “Witr, between ‘Isha’ prayer and the break of dawn.” And there are many ahaadeeth on this topic, which indicates that Witr ends with the break of dawn. 

If a worshipper does not know when dawn is, he may rely on a muezzin who is known to pay attention to the right time. If the muezzin who pays attention to the right time gives the adhaan, then he has missed witr. As for the one who gives the adhaan before dawn breaks, he has not missed witr when he gives the adhaan and this does not mean that it has become haraam for one who is fasting to eat and drink, and the time for Fajr prayer has not begun with this adhaan. The Prophet (peace and blessings of Allaah be upon him) said: “Bilaal gives the adhaan at night, so eat and drink until Ibn Umm Maktoom gives the adhaan.” Saheeh – agreed upon. Ibn Umm Maktoom was a blind man who did not give the adhaan until he was told that dawn had come. From what we have mentioned it is clear that the time for witr ends with the first adhaan if the muezzin pays attention to the correct time of dawn, but if the muezzin gives the adhaan when the Muslim is in the final rak'ah of witr, he should complete it because he cannot be certain that dawn has come just from hearing the adhaan, and there is nothing wrong with that in sha Allaah. 

If a person misses Witr, it is prescribed for him to pray what he usually prays during the day, but he should make it even-numbered by adding one rak’ah. If he usually prays three, he should make it four, and if he usually prays five, he should make it six, and so on, saying the tasleem after each two rak’ahs. It is narrated in Saheeh Muslim that ‘Aa’ishah  (may Allaah be pleased with her) said: If the Prophet (peace and blessings of Allaah be upon him) missed witr at night because he was sick or sleeping, he would pray twelve rak’ahs during the day. And he usually prayed eleven rak’ahs, but if sickness or sleep kept him from doing that, he would pray twelve rak’ahs as ‘Aa’ishah (may Allaah be pleased with her) said, saying the tasleem after each two rak’ahs, as ‘Aa’ishah (may Allaah be pleased with her) said: “The Messenger of Allaah (peace and blessings of Allaah be upon him) used to pray ten rak’ahs at night, saying the tasleem after each two rak’ahs, and praying witr with one.” Saheeh – agreed upon. And the Prophet (peace and blessings of Allaah be upon him) said: “The prayers of the night and day are two by two.” Narrated by Imam Ahmad and the authors of al-Sunan with a saheeh isnaad from the hadeeth of Ibn ‘Umar (may Allaah be pleased with him). It is narrated in al-Saheehayn with the wording, “The night prayers are two by two,” as we stated at the beginning. And Allaah is the Source of strength. End quote. 

Majmoo’ Fataawa Ibn Baaz, 11/305-308 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked: I am keen to offer witr prayer at the best time for it, before dawn breaks, but sometimes I cannot do it before dawn. Is it permissible for me to pray witr after dawn breaks? 

He replied: 

If dawn breaks and you have not prayed witr, then do not pray witr, rather pray four rak'ahs during the day if you usually pray witr with three, and six if you usually pray witr with five, and so on. 

Because if the Prophet (peace and blessings of Allaah be upon him) missed the night prayers, he would pray twelve rak’ahs during the day. End quote. 

Majmoo’ Fataawa Ibn ‘Uthaymeen, 14/114 

It was narrated from a number of the Sahaabah that there is nothing wrong with praying witr after the adhaan of Fajr until the iqaamah is given – such as Ibn Mas’ood (as was narrated by al-Nasaa’i (1667) and classed as saheeh by al-Albaani in Saheeh al-Nasaa’i), Ibn ‘Abbaas (as narrated by Maalik in al-Muwatta’ (255)) and ‘Ubaadah ibn al-Saamit (as narrated by Maalik in al-Muwatta’ (257)). 

Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) was asked about a person who sleeps and misses Witr prayer. 

He replied: He may pray between dawn and Fajr prayer, as ‘Abd-Allaah ibn ‘Umar, ‘Aa’ishah and others did. Abu Dawood narrated in his Sunan that Abu Sa’eed said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Whoever sleeps and misses Witr or forgets it, let him pray it when morning comes or he remembers.” There are different reports from Ahmad as to whether he should make up the even-numbered rak’ahs too. The correct view is that he should make up even-numbered rak’ahs too. It is narrated in a saheeh report that the Prophet (peace and blessings of Allaah be upon him) said: “Whoever sleeps and misses a prayer or forgets it, let him offer it when he remembers it, for that is the time for it.” This includes obligatory prayers, qiyaam al-layl, Witr and regular Sunnah prayers.” End quote from al-Fataawa al-Kubra, 2/240. 

If the Muslim does either of these two things, there is no sin on him in sha Allaah. 

And Allaah knows best.