Is it considered disbelief to return to sleep—after waking up and a short time before the prayer's deadline, a time insufficient for ablution and prayer due to obsessive doubts (waswas)—?

Disbelief does not affect the questioner. It appears she is suffering from misgivings (waswas) concerning purity and matters of disbelief. She must rid herself of these misgivings by seeking refuge in Allah from them and desisting from them. She should increase her supplication, turn to Allah, remember Allah, and occupy herself with what benefits her in religious and worldly matters. Among the blameworthy misgivings that cause the waste of prayer time are those concerning purity, and these must be cured by ignoring them. If the questioner addresses these misgivings, the time mentioned in the question will be sufficient for purification and performing the prayer on time, thereby removing the sin of delaying the prayer. Sleeping times must be organized to avoid conflict with prayer times.