If a man still had some days to make up from the month of Ramadan and he did not make them up before the start of the next Ramadan, what should he do?
It is well known that Allaah, the Most Glorified, the Most High says:
So whoever of you sights (the crescent on the first night of) the month (of Ramadân i.e. is present at his home), he must observe Saum (fasts) that month, and whoever is ill or on a journey, the same number [of days which one did not observe Saum (fasts) must be made up] from other days. [Al-Baqarah 2:185]
So, this man, who broke his fast for some valid legal reason, is obliged to make up for it, in obedience to the Command of Allaah, the Most Glorified, the Most High. He must make it up in that year, and he should not delay it until after the following Ramadan. ‘Aishah, may Allaah be pleased with her, said:
”I had days to make up from Ramadan, and I was unable to make them up until Sha’ban (i.e. the month before the following Ramadan).” 
This is due to the Messenger of Allaah’s relationship with her, so her saying ’I was unable to make them up until Sha’ban’ is a proof that they must be made up for before the start of the next Ramadan.
But if he delayed it after the following Ramadan, he should seek forgiveness from Allaah and repent to Him, and be remorseful for what he has done, and make up for these days. The obligation to make it up for is not removed merely because it has been delayed, so he should make up for these days even if it was after the next Ramadan. And Allaah is the Granter of Success.
 Reported by Al-Bukhari in the Book of Fasting, in the Chapter: When the Missed Days of Fasting Should be Made up for (1950).
Shaykh Muhammad bin Saalih al-`Uthaymeen
Fatawa Arkaanul-Islaam, DARUSSALAM, Vol.2, P.673.