Having lived all my 65 years in various parts of Britain (South Wales, Cambridge, Oxford, London and Brighton) I'd say that the normal British pronunciation is "ishoo", though the more phonetic "issyoo" is a universally accepted variant. I think my own pronunciation started as "ishoo", veered towards "issyoo" and then edged back towards the majority version.
Cambridge Dictionaries online pronunciation guide gives only "ishoo" as the British pronunciation, but perhaps they are not set up in a way that allows alternative minority versions.
I think that "issyoo" has its main stronghold among politicians, and in the broadcast legacy media such as the BBC and C4. The British social media site The Student Room has a thread that pretty explicitly confirms this (before degenerating into silliness). The small minority of contributors who used the more phonetic version appealed to the authority of usage among politicians.
Also, the estimable vlogger Sargon of Akkad, who in my opinion has a beautiful educated British-English accent, once prefixed one of his YouTube videos with a note about this topic. He said that he had recently noticed himself moving from "ishoo" to "issyoo", and put it down to listening to too many politicians. He promised to try to resist the affectation in future.