Is it permissible to perform tayammum with a stone for janabah if water is unavailable and one is unable to perform ghusl before Fajr due to illness, while still being able to perform wudu', or is it obligatory to delay the Fajr prayer until ghusl is performed?

If you are harmed by using water due to the cold, you must take all possible means to perform ghusl without harm, such as heating the water or performing ghusl in a warm place. It is not permissible to perform tayammum as long as you can use water by these methods.

If you are unable to use water at all, and it harms you no matter what you do, but you can wash some limbs, then wash what you can and perform tayammum for the rest, and do not delay the prayer beyond its time. This is the view of the Shafi'is and Hanbalis. Abu Hanifa and Malik differed on some details. It is permissible to perform tayammum on a stone according to the Malikis and Hanafis.