When do the sick person and one without water perform tayammum?

Answer (and Allah is the One who grants success): It has been said so, and this is what the later scholars of the school determined. Others—both within the school and outside it—said: If the sick person has no hope of his ailment disappearing within the prayer time, he may perform tayammum and pray whenever he wishes within that time; but if he expects the ailment to be removed at the end of the time, he waits to pray until the end of the time. This school of thought is closer to truth and correctness, because saying that it is obligatory to delay the prayer absolutely entails:
1. imposing severity and constraint upon the sick person—which contravenes ease;
2. the sick person may fall asleep at the end of the time—especially before Fajr—and thus miss the prayer;
3. his illness may worsen and thus he misses the prayer.
All of that conflicts with the obligation to maintain the prayers, known from His saying: “Maintain with care the [obligatory] prayers and [in particular] the middle prayer.” [Al-Baqarah:238]
Moreover, praying at sunset is strongly disliked; so it is not fitting to delay the prayer to that time.
Yes: one who lacks water—if he thinks that by delaying his prayer he will find water—should wait to pray, but not so long as to enter a time in which the Messenger of Allah (May Allah bless him and his family and grant them peace) forbade praying. If the one lacking water despairs of finding it, he may pray whenever he wishes within the time.
Source : Min Thimār al-ʿIlm wa al-Ḥikmah vol.1