Praise be to Allah.
Firstly: the more correct scholarly view is that rinsing the mouth and nose are obligatory in both wudoo’ and ghusl. This has been explained previously in the answer to question no. 88066 .
Secondly: if a man or woman does ghusl to remove major impurity (such as janaabah or menses), this ghusl counts for wudoo’ too.
Muslim (327) narrated from Jubayr ibn Mut‘im (may Allah be pleased with him) that mention of ghusl for janaabah was made in the presence of the Prophet (blessings and peace of Allah be upon him) and he said: “As for me, I pour water over my head three times.”
Al-Bayhaqi (may Allah have mercy on him) included this report in a chapter entitled (1/63): Chapter on the evidence for wudoo’ being included in ghusl.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
If a person becomes junub, doing ghusl suffices for wudoo’ too, but it is essential to rinse the mouth and nose. End quote.
Fataawa Noor ‘ala ad-Darb (120/20).
Thirdly: the scholars differed as to whether it is obligatory to wash the parts of the body consecutively in ghusl.
It says in al-Mawsoo‘ah al-Fiqhiyyah (11/200, 102):
Washing the parts of the body in order and consecutively when doing ghusl is not obligatory according to the majority of fuqaha’.
Al-Layth said: It is not essential to wash them consecutively. This is a different view from that of Imam Maalik; the preferred view according to his companions is that it is obligatory to do them consecutively, and there is a similar view among the companions of Imam ash-Shaafa‘i.
But according to the view of the majority, if someone does wudoo’ with ghusl, he does not have to wash the parts of the body in order when doing wudoo’.
Because of that, if he omits to wash one part or a spot of one part, whether it is the parts that are to be washed in wudoo’ or not, he may wash the part that was omitted on its own afterwards, regardless of whether the time interval is long or short, and if he has washed his body except the parts that are washed in wudoo’, he can wash them now, and he does not have to wash them in order.
Because of that, ash-Shaafa‘i said: If he omits wudoo’, or rinsing the mouth or nose, when doing ghusl, it is disliked (makrooh), and it is recommended (mustahabb) for him to do those actions, even if a long interval has elapsed, without having to repeat ghusl. However, according to the Hanafis and Hanbalis, he must make up these actions, because they are obligatory in ghusl according to them. But in the case of wudoo’, rinsing the mouth and nose is Sunnah according to the Hanafis, and is not obligatory.
Al-Mawsoo‘ah al-Fiqhiyyah (11/100-101).
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked:
If, after doing ghusl for janaabah or menses, a woman remembers that she has nail polish on her nails, does she have to repeat her ghusl?
He replied: According to the madhhab, she does not have to do that, because doing the actions consecutively is not a condition of ghusl being valid, in their view. But what we think is that it is a condition. How can we regard it as a condition in wudoo’ but not in ghusl? So she must repeat ghusl. End quote.
Thamaraat at-Tadween (p. 21).
What appears to be the case is that your ghusl is valid, even according to the view that the actions in ghusl must be done consecutively, because the time that elapsed between ghusl and rinsing the nose was short, and may not have been more than two or three minutes. Such a short time does not cancel out doing the actions consecutively.
Fourthly: this whiteness that appears on the teeth is tartar that accumulates on the teeth because of not taking proper care of them or regularly cleaning them with the siwaak or toothpaste.
The scholars of the Permanent Committee for Ifta’ were asked: When we eat, we get some bits of food between our teeth, then when we do wudoo’ or ghusl, we cannot remove these bits. Is our wudoo’ and ghusl valid?
Wudoo’ and ghusl are valid even if any bits of food remain between the teeth, but removing them is better. End quote.
Fataawa al-Lajnah ad-Daa’imah (5/234).
Based on that, your wudoo’ is valid, but you should regularly clean your teeth and take care of them.
And Allah knows best.