If he does ghusl and intends to remove two types of impurity, when should he rinse his mouth and nose?

Praise be to Allah.

Firstly: 

Ghusl does not take the place of Wudu unless ghusl is done to remove major impurity (janaabah, menses, nifaas). Shaykh Ibn Baz (may Allah have mercy on him) said:  

If ghusl is being done to remove janaabah, and the person who is doing ghusl intends to remove both types of impurity, minor and major, it is sufficient for both. But if the ghusl is done for some other reason, such as ghusl for Jumu’ah, or bathing to cool oneself down or clean oneself, then it does not take the place of Wudu even if one intends that, because when bathing for these purposes, one does not pay attention to the order in which the parts of the body are washed, which is one of the obligatory parts of Wudu, and because there is no major purification in the intention whereby the minor purification could be included in the intention, as in the case of ghusl to remove janaabah."(Majmoo’ Fatawa Ibn Baz  10/173). 

See also the answer to question no. 82759. 

Secondly: 

Rinsing the mouth and nose is essential when doing ghusl, as is the view of the Hanafis and Hanbalis. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: Among the scholars are those who say that ghusl is not valid without them (rinsing the mouth and nose), like Wudu. 

And some said that it may be valid without them. 

The correct view is the former, because Allah says (interpretation of the meaning): “purify yourselves” [Al-Maa'idah 5:6], which includes the entire body. The inside of the mouth and nose are part of the body which must be purified. Hence the Prophet (peace and blessings of Allah be upon him) ordered that they be rinsed in Wudu because they are included in the verse: “wash your faces” [Al-Maa’idah 5:6]. 

If they are included in washing the face, which is something that must be cleansed during Wudu, then it is more likely that they are included in ghusl, because purification in the case of ghusl is more comprehensive."(Al-Sharh al-Mumti’).  

Thirdly: 

It is mustahabb for the person who is doing ghusl to do Wudu at the beginning of his ghusl, but if he does not do that and he pours water over his entire body, that is sufficient and his ghusl is valid. He may rinse his mouth and nose at the beginning of ghusl or at the end or in the middle, because doing things in a certain order is not required in ghusl. 

See al-Mughni (1/140). 

And Allah knows best.