One factor that does still seem current is that local people at the lower end of the social scale tend to omit the /r/ in ‘Shrew-’ and pronounce the name /ʃuːzbri/. This is mentioned in the Talk page (under ‘Pronunciation') of the WP Shrewsbury article here.
Recently I have had comments, when I have used the /əʊ/ pronunciation, from people who find my version funny. When I protest that I grew up there, I find that that doesn’t cut much ice, and I’m told that “nobody says /ʃrəʊzbri/ any more”. I realize that the situation is quite similar to that of Marlborough. Traditionally, the pronunciation of this town in Wiltshire is /mɔːlbrə/. If you look at the Wikipedia article on Marlborough you will find the two pronunciations /mɔːlbrə/ and /mɑːlbrə/. As long as I can remember, I have known that the “correct” pronunciation is the one with /ɔː/, yet I have always felt reluctant to use that because it felt to me old-fashioned and posh. I realize that /ʃrəʊzbri/ has become the old-fashioned and posh pronunciation, too.