What is the ruling on delaying the Dhuhr prayer until a quarter of an hour before the 'Asr prayer due to an inability to wake up from sleep, even while hearing the alarm?

Delaying the Dhuhr prayer when it is not hot is considered abandoning what is mustahabb (recommended), as it is recommended to perform it at the beginning of its time. This is what the scholars have chosen, and no disagreement is known concerning it, due to what has been authentically reported from the Prophet, peace and blessings be upon him, and from Aisha, may Allah be pleased with her, that she never saw anyone more prompt in performing Dhuhr than him, or than Abu Bakr and Umar.

Abandoning what is mustahabb, at the very least, is contrary to what is most preferable, and it may be makrooh (disliked) if there is evidence for its dislike. Therefore, one should be keen to perform the prayer at the beginning of its time. However, if the prayer is delayed for any reason and you pray it at the end of its time before it expires, there is no harm upon you.