Is it permissible to perform tayammum instead of wudu for someone who suffers from angina and catches a cold despite performing wudu with warm water in a closed space, given the futility of visiting a doctor?

It is permissible to perform tayammum when water is unavailable or when one is unable to use it. Among the excuses that permit it is the fear of contracting an illness, even if it is a minor one, or if the fear is of a frightening illness. For this fear to be valid, it is sufficient to think that an illness will occur, or worsen, or that recovery will be delayed, and not only to fear perishing. So, if you fear contracting an illness upon using water, tayammum is permissible for you. If you are able to wash some limbs without harm, you must wash them, then perform tayammum for the remaining ones, observing the proper order and sequence. However, if the fear is merely an illusion, then wudu' is obligatory.