I have applied to five different universities. Four of them are in the UK and the last one elsewhere in Europe. Based on initial discussions, two of the UK universities consider me a competitive applicant (they encouraged me to apply after informal inquiries). Given my impressive publication record, I think I am the best applicant for the position which is not located in the UK.
Most probably, I will get an offer from the non-UK university quite soon. However, I would not like to accept the offer before hearing anything from the UK universities. Suppose I am also the best candidate for the UK institutions. How fast will they decide? The places are single-slot, so would it be rude to ask one week after the application period closes?
I would only email the UK institutions if the non-UK institution says yes, and write something along the lines
Sorry for contacting you so soon after application period has ended, but I have been offered a position in [some place] already at this time. I understand that you probably have received many good applications, and may not decide yet who to offer the position first. However, I thought you might be able to tell if others' applications were much stronger than mine. I would prefer to study with you, which is the reason I am sending this email.
I would then decline the non-UK offer only if the response was very positive sounding one.