Is it permissible to make up all delayed Fajr prayers for four years first, then begin making up delayed Dhuhr prayers for the same period, and so on until all missed prayers are completed?

Scholars have differed regarding the obligation of observing order when making up missed prayers. The Hanafis maintained its obligation if the missed prayers were few, while the Malikis and Hanbalis held it to be obligatory in all cases. The Shafi'is, however, considered it recommended. Therefore, if not observing the order makes it easier for you to make up the missed prayers, there is no harm in doing so, acting upon the Shafi'i madhhab. Nevertheless, it is recommended to make them up in order, following the Prophet, may Allah's peace and blessings be upon him.