Does a prayer room designated for Muslims in British universities—which they do not own and is under the university's administration—take the ruling of a mosque, and is it permissible to perform i'tikaf in it?

So long as the aforementioned place has not exited the ownership of the university, and has not reached the stage of a legally recognized waqf (endowment), then the rulings pertaining to a mosque, such as i'tikaf (seclusion for worship) and the like, do not apply to it. Ibn al-Arabi said: "If a man were to build a mosque within his house, and restrict it from people, reserving it for himself, it would remain his property and would not attain the status of a mosque."