What is the maximum permissible delay for making up a missed prayer due to an excuse such as sleep or the like, and is it permissible to say that the allowed time for delay extends until just before the time of the next prayer, based on the hadith: "There is no negligence in sleep; rather, negligence is in wakefulness, when one delays a prayer until the time of another prayer enters"?

The question is unclear. However, if you are asking about the permissible limit for delaying a prayer for someone excused by sleep or forgetfulness, the answer is that it is not time-bound. Such a person is not held accountable until their excuse is removed, as the Prophet (peace be upon him) said: "Whoever forgets a prayer, let him pray it when he remembers it; there is no expiation for it except that." And in a narration by Muslim: "Whoever forgets a prayer or sleeps through it, its expiation is to pray it when he remembers it."