A wife has Ramadan fasts to make up. Should she fast the six days of Shawwal first, or make up first?

It is better to make up first because it is obligatory; but according to the view held by many Shafi'i scholars, fasting six days of Shawwal before making up is still valid, and some allow combining the intention of making up with fasting on days of Shawwal to gain the recommended reward.
The Prophet ﷺ said that whoever fasts Ramadan and follows it with six days of Shawwal, it is as if he fasted the whole year (Sahih Muslim, Kitab as-Siyam). Some scholars hold that the full reward is only for those who complete Ramadan first, so making up comes first. For women with many days to make up because of postnatal bleeding or breastfeeding, a practical option is to make up in Shawwal with the intention of making up; she can still hope for the virtue of Shawwal. What matters is that making up is not delayed beyond the next Ramadan. See the question on delayed making up.