What is the ruling on a young man who relieves himself in a bucket with his back to his father, and asks his father to leave so that he can perform istinja' (washing after defecation), wudu' (ablution), or ghusl (ritual bath) for janaba (major ritual impurity); fearing that he will miss the prayer or lose its reward, despite the difficult circumstances they are living in?

What you are doing is correct and you will be rewarded for it, for the greater the hardship, the greater the reward. However, if you fear illness from bathing, you have a concession to perform tayammum until the cold subsides. If you fear harm, wash what you can, then perform tayammum for the rest. Tayammum is permissible due to fear of harm from using water, such as an injury or severe cold, based on the Almighty's saying: {And do not kill yourselves} and the narration of 'Amr ibn al-'As. It is sufficient for you to fear a cold or illness, so you perform tayammum after washing what you can wash without harm, and your prayer is valid with tayammum, and you are not considered to have abandoned the obligation.