Which takes precedence: making up missed Ramadan fasts or fasting in expiation for breaking an oath or fulfilment of a vow?
Question

I have some fasts to make up from Ramadan, and I also have some fasts to observe in expiation for breaking an oath (kafaarat yameen). I heard once that the first thing I should do is make up the missed Ramadan fasts, and after that I should observe the expiatory fasts. Is doing it in that order obligatory, or is it permissible to do it differently?
Praise be to Allah.If someone owes days from Ramadan, he may delay making them up, so long as the next Ramadan has not begun.
Ibn Qudaamah (may Allah have mercy on him) said: To sum up, if someone owns a fast from Ramadan, then he may delay it so long as the next Ramadan has not begun, because of the report narrated by ‘Aa’ishah, who said: I used to have fasts that I still owed from Ramadan, and I would not make them up until Sha‘baan. Agreed upon. It is not permissible for him to delay making them up until the next Ramadan begins without a valid reason, because ‘Aa’ishah (may Allah be pleased with her) did not delay making up missed fasts to that extent; if she could have, she would have delayed hem.
End quote from al-Mughni (3/85)
With regard to expiation for breaking an oath (kafaarat yameen), the scholars differed as to whether it is obligatory to do it immediately or whether it may be deferred.
It says in al-Mawsoo‘ah al-Fiqhiyyah (10/14): The majority of scholars are of the view that it is not permissible to delay expiation for breaking an oath, and that when the oath is broken, the expiation must be offered immediately. End quote.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: Part of heeding the solemnity of oaths is offering expiation immediately after breaking the oath; the expiation becomes obligatory with immediate effect, because the basic principle with regard to obligatory matters is that they must be done immediately.
End quote from al-Qawl al-Mufeed ‘ala Kitaab at-Tawheed (2/456). See also ash-Sharh al-Mumti‘ (15/159).
The Shaafa‘is, according to the more correct opinion, are of the view that it is obligatory to offer the expiation immediately if breaking the oath involves an act of disobedience, such as if a person swore to refrain from committing a sin, then he does it. They said: In this case, he must offer expiation immediately.
An-Nawawi (may Allah have mercy on him) said: As for expiation, if it has to do with an act that did not involve deliberate transgression, such as expiation for accidental killing, or expiation for breaking an oath, in some instances, then it may be done later on, and there is no difference of scholarly opinion in this case, because the act is excused.
But if the act involved deliberate transgression, then must the expiation be offered immediately or can it be deferred? There are two opinions which were both narrated by al-Qaffaal and his companions; the more sound view is that it must be offered immediately.
End quote from al-Majmoo‘ (3/70)
So according to the majority view, expiation for breaking an oath should be given precedence, because it must be offered immediately, whereas making up missed Ramadan fasts may be deferred.
If the time is short and there are only a few days left until Ramadan, so there is not enough time to both make up the missed fasts and offer expiation, then precedence should be given to making up the missed fasts, because that is more important, and the scholars stated that it should be given precedence over fasting in fulfilment of a vow.
An-Nawawi (may Allah have mercy on him) said: If he missed any of the fasts in Ramadan because of a valid reason, and the excuse is no longer applicable, then he must make up the missed Ramadan fasts, because that is more important than fasting in fulfilment of a vow.
End quote from al-Majmoo‘ (6/391)
And Allah knows best.