What should a person do if they performed tayammum and prayed Fajr due to extreme cold, and then the excuse was removed? Is their prayer valid, or must it be repeated? And what are the implications if they are unable to perform ghusl until after the time for the prayer has elapsed?

If you are harmed by using water due to cold, you must heat the water or bathe in a warm place, and tayammum is not permissible as long as you are able to do so.

If you are harmed by using water and are unable to bathe, perform tayammum and pray. Then, if you become able to use water, you are not required to make up the prayer. This is the more preponderant opinion.

However, if you are unable to bathe and the time for prayer expires, you are obligated to make up that prayer with tayammum if the inability to use water persists, or by bathing if you become able to use water. There is no sin upon you if you were ignorant or forgetful.