Sheikh ibn Uthamin, may Allah have mercy upon him, was asked about a woman who had a debt (missed days) from the month of Ramadan, is it permissible for her to fast the six days of Shawwal before making up the days or should she make up the days first?
The Sheikh answered by saying: If the woman has days to make up from the month of Ramadan then she can not fast the six days of Shawwal until she has made these days up.
This is because the Prophet, peace and blessings of Allah are upon him, said, Whoever fasted the month of Ramadan and then followed it by fasting six days from Shawwal (Then this is equivalent to fasting the entire year) Therefore the one, who has to make up days from Ramadan, has not fasted the month of Ramadan so she will not receive the reward for the six days of Shawwal until she makes up the missed days from Ramadan.
And if making up the missed days of Ramadan were to take the entire month of Shawwal; for example the woman experienced post child-birth bleeding and as a result she did not fast a single day in Ramadan; and then she began to make up the days of Ramadan in Shawwal and she did not complete the days until the month of Dhu al-Qi'dah entered, then at this time she can fast the six days of Shawwal and she will be rewarded for her fasting the days of Shawwal; and this is because her delay was due to necessity. Meaning her delaying the six days of Shawwal is excused, therefore she will be rewarded.