Is it permissible to delay the completion of the ritual bath (ghusl) from major impurity (janabah) – after washing the private parts in the bathroom – until reaching a warm room? And is it valid to perform tayammum and pray if one is unable to perform ghusl due to cold weather or work circumstances? And what is the ruling on prayers performed with such tayammum?

There is no harm in completing the ghusl (definition) elsewhere, and continuity in ghusl is not obligatory according to the most sound opinion. Tayammum (definition) is not permissible due to cold unless there is a fear that using water will cause illness. Your work with your father and brothers is not an excuse that permits tayammum while you are able to perform wudu and ghusl. The command of Allah must be given precedence, and ghusl must be performed. If you prayed with tayammum while able to perform ghusl, you must make up those prayers.