Does the ruling of declaring one who intentionally misses a prayer until its time has passed an apostate apply to someone who woke up lazily after Fajr prayer time had elapsed, then prayed immediately upon waking up?

Whoever misses a prayer due to sleep or forgetfulness until its time has expired is not sinful. Scholars have differed on the immediacy of making up the prayer (qada'): The opinion of the majority: It is obligatory to make it up immediately, and one sins by delaying it. The Shafi'i madhhab: It is permissible to delay making it up because it is on a relaxed basis ( 'ala al-tarakhi), unless there was negligence.

As for one who intentionally delays the prayer until its time has expired, the correct view according to the Shafi'is is that making it up must be done immediately. It is safer to hasten to make up the missed prayer and not delay it.