Is it sufficient to make up missed prayers with ablution (wudu') in a state of major ritual impurity (janabah) when unable to perform a full ritual bath (ghusl) due to extreme cold and a weakened immune system, or is it obligatory to wait for the ritual bath upon returning from work?

It was obligatory for you to heat the water for wudu’ if possible. And if you feared that using it would cause illness, or worsen an existing one, or delay healing, then you must perform tayammum (definition) for the prayers that you perform outside the home, until you are able to use water. There is also a saying among some scholars that your wudu’ is sufficient until you are able to perform ghusl (definition).