What should one do if they experience a wet dream in their lodging in a non-Islamic country where there is only one shared bathroom for males and females at specific times, and the males' time begins after sunrise? Should they perform tayammum and pray, or wait for the bathroom time to perform ghusl and make up the Fajr prayer?

If a person in a state of major ritual impurity (junub) wakes up and realizes they are unable to use water until after the time for prayer has elapsed, they should perform tayammum (dry ablution) and pray, without delaying the prayer. There is no need for them to repeat it.

Imam An-Nawawi said: If a crowd gathers at a well and it's impossible to draw water except by taking turns, and you know that your turn will not come until after the time for prayer has passed, then the more apparent ruling is that one should pray within the time by performing tayammum.

In "Hashiyat Al-Rawd," it states: For someone who can go to the حمام (bathhouse) but cannot leave it until after the time has elapsed, such as a woman with her children, the more apparent ruling is that she should perform tayammum and pray outside the حمام.