Worcester is pronounced like Wooster. Like rooster but with a W at the beginning. Shire isn't pronounced like the Shire in Middle Earth, but like New Hampshire, so it sounds more like "sure". So the whole thing is pronounced like wooster-sure. It's really easy once you stop trying to force the R and the C in there as many people do.
I agree about the 'oo', I just put that as an example of a word people can relate to.
Not so much about the 'sheer' part though. The most correct someone with an American accent can get is imo like 'sure', but if you're British you just don't pronounce the R at all so I guess with a British accent it could sound a bit like 'sheer', definitely not with an American accent though.
I said the word 'shire' by itself, sounds like fire. But the suffix '-shire' which is added to make County names of Britain is pronounced exactly like sheer
Not so much about the 'sheer' part though. The most correct someone with an American accent can get is imo like 'sure', but if you're British you just don't pronounce the R at all so I guess with a British accent it could sound a bit like 'sheer', definitely not with an American accent though.