A British pronunciation issue
Solution 1:
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.
Solution 2:
J. C. Wells in his Longman Pronunciation Dictionary (3rd edition, 2008) has /ˈɪʃu:/ as the only American pronunciation of the word, and /ˈɪsju:/, /ˈɪʃju:/ and /ˈɪʃu:/ for British English, with the following distribution:
/ˈɪsju:/: 30%
/ˈɪʃju:/: 21%
/ˈɪʃu:/: 49%
Further information is provided about prevalence according to age, with 40% of older speakers and more than 90% of younger speakers using one of the /ʃ/ variants, /ˈɪʃju:/ or /ˈɪʃu:/. The conclusion from the figures is that the /ˈɪsju:/ pronunciation is being replaced in British English by the other ones.
/ʃ/ is the sound that is found in shoe, /j/ in you.
Solution 3:
I've lived on both sides of the pond, and in Old Blighty I don't remember ever hearing anything other than "iss-you" -- not that nobody ever did say it the other way, mind, that's just the only place I've heard it said that way. In Yankworld I've always heard (and said) "iss-shoe" and not the other way.
This is not to say that in certain places in the UK you might hear "iss-shoe" and in the US "iss-you".
I remember being made lightly fun of at Cheltenham Grammar School for pronouncing "tube" as "toob" rather than "tee-oob". I tried to get them to show me where there was a "y" in that word, but in vain (you know, "tyoob"). Nobody wanted to consider my position. Eventually I decided "when in Rome" and all that. Coming back to the US I had to unlearn a number of pronunciation issues. After 40 years I still have a hard time pronouncing "headmaster" like a Yank. My mouth absolutely wants to say "headMAHster" with the long "a" and emphasis on "mas" -- instead of the American short "a" with emphasis on "head".