The scholars differed concerning the ruling on rinsing the mouth and nose in wudoo’ and ghusl. The correct scholarly view is that these actions are obligatory, and wudoo’ or ghusl is not valid without them, because they are included in washing the face which is enjoined in the Qur’aanic verse. 

Al-Hajaawi said in al-Zad, in Baab Furood al-Wudoo’ wa Sifatuhu (The obligatory parts of wudoo’ and its description), p. 29: 

The obligatory parts (of wudoo’) are six: Washing the face – of which the mouth and nose are part; washing the hands; wiping the head – of which the ears are part; washing the feet up to the ankles; doing these actions in order; and doing them continuously, without interruption, which means not delaying the washing of one part until the part before it has dried. End quote. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said in al-Sharh: 

The words “of which the mouth and nose are part” mean they are part of the face, because they are located on the face, so they are included with it. Based on this, rinsing the mouth and nose are among the obligatory parts of wudoo’, but they are not separate (rather they are included with the face). This is similar to the words of the Prophet (blessings and peace of Allah be upon him): “I have been commanded to prostrate on seven bones: the forehead – and he pointed to his nose…”  . End quote. 

Al-Sharh al-Mumti‘, 1/119 

The scholars of the Standing Committee for Issuing Fatwas said: 

Rinsing the mouth and nose in wudoo’ is proven from the actions and words of the Prophet (blessings and peace of Allah be upon him) and they are included in washing the face. So wudoo’ is not valid if they are omitted or if one of them is omitted. End quote. 

Fataawa al-Lajnah al-Daa’imah, 4/78 

See also al-Mughni by Ibn Qudaamah, 1/83 

The fact that the verse does not explicitly mention rinsing the mouth and nose does not mean that they are not obligatory, because the Sunnah explains the Qur’aan, and the Sunnah teaches us that we should rinse the mouth and nose. It is not narrated from the Prophet (blessings and peace of Allah be upon him) that he omitted one or both of them in wudoo’. This is an explanation of the command mentioned in the Qur’aan to wash the face when purifying oneself. 

So if a person omits rinsing the mouth and nose when purifying himself, his purification is not valid, whether he did that deliberately or by mistake. 

The scholars of the Standing Committee for Issuing Fatwas said: 

If a person forgets to wash any of the parts of the body that should be washed when doing wudoo’, or he omits any portion of that, even if it is small, if it was during wudoo’ or immediately afterwards and the traces of water are still on his limbs and have not dried yet, then he should wash the part that he forgot and those that come after it only. But if he remembers that he forgot to wash one of the parts of the body that are to be washed in wudoo’, or a portion of it, after his limbs have dried or during the prayer or after finishing the prayer, then he has to do wudoo’ all over again, just as Allah has prescribed repeating the prayer in full. Allah has enjoined washing all the parts of the body that are to be washed in wudoo’, so if a person omits part of that, even if it is a small area on one of the parts that are to be washed in wudoo’, it is as if he forgot to wash all of it. This is what is indicated in the report narrated by ‘Umar ibn al-Khattab (may Allah be pleased with him) who said: The Messenger of Allah (blessings and peace of Allah be upon him) saw a man doing wudoo’, and he omitted a spot the size of a fingernail on his foot. He told him to repeat his wudoo’ and his prayer, and he went back and prayed again.

Narrated by Muslim, 243; Ibn Maajah, 666. End quote.  

Fataawa al-Lajnah al-Daa’imah, 4/92 

For more information please see the answer to question no. 149908 

And Allah knows best.