Is it permissible for a university student who missed previous prayers to begin observing the prayers of the present day as they become due and make them up immediately, while delaying the making up of prayers missed over the past two months until a suitable time becomes available for that?

It is not permissible to delay the prayer beyond its appointed time due to studying. Prayer holds a great status in Islam. Prayer can be performed anywhere if the mosque is closed. If a prayer is missed, it must be made up immediately. The order between missed prayers is recommended but not a condition. All missed prayers must be made up immediately and as much as possible, with a minimum of two days' worth of prayers per day.