[How should one act when time is tight and several obligations conflict?]

Answer: A Muslim ought not delay the prayer if he expects tasks to pile up near the end of the time. If he does delay until the end and a case like the question arises, then aiding the patient takes precedence if the illness is severe and there is fear of death or grave harm if help is not immediate—such as hemorrhage. If he fears missing the prayer while giving aid, he should pray however he can: perform tayammum, gesture for bowing and prostration, and pray even not facing the qiblah: “So fear Allah as much as you are able.” [At-Taghābun:16]
The proof is His saying, Exalted is He: “Maintain with care the [obligatory] prayers and [in particular] the middle prayer, and stand before Allah devoutly obedient. And if you fear (an enemy), then [pray] on foot or riding.” [Al-Baqarah:238–239] Allah, Exalted is He, permitted in this verse that when believers are arrayed against an enemy, they pray however possible—this is called the prayer of engagement (ṣalāt al-musāyafah). The fighter prays mounted on his horse while fighting, or on foot while engaged, gesturing for bowing and prostration; turning, moving, striking, etc., do not harm the prayer.
Thus the verse indicates that preserving life takes precedence over some obligations; accordingly, preserving the life of a believing patient takes precedence over some obligations of the prayer—except that nothing of the prayer’s obligations that can be performed alongside aiding the patient should be dropped, as noted above.
Source : Min Thimār al-ʿIlm wa al-Ḥikmah vol.1