'Enquire', and the associated noun 'enquiry', are more common in British English, while 'inquire' and 'inquiry' are more common in American English. In Australia, we use either spelling although enquire and enquiry for the general sense of 'ask', and inquire and inquiry for a formal investigation, is preferred.

Is it an inquiry or enquiry?

In the US, inquiry and enquiry are interchangeable, but be aware that many will consider enquiry to be a spelling mistake of inquiry. In the UK, inquiry and enquiry are interchangeable. However, it is becoming preferable to use inquiry to denote an investigation and enquiry to denote a question.

Is it enquiry or inquiry in the UK?

There is a very simple answer here - there is no difference in meaning. The spelling with 'e' is British, the spelling with 'i' is North American. The same goes for the nouns, 'inquiry' and 'enquiry'. There are of course other differences in spelling between American and British English.