No, they don't. Most English speakers (including the whole of North America) would not sound the 'g' in singer but would in finger. However, it seems I'm a little wrong about exactly
where they do rhyme. My dad, from Bradford, does it, but according to this map of ng-coalescence he probably got it from his mum (from Bolton, or it was different when he was born 80-odd years ago). It does put Manchester right in the uncoalesced ng area though which does stretch right down to Birmingham.
View attachment 441217
from
Phonological history of English consonant clusters - Wikipedia
It rather annoyingly doesn't include Wales (
Rebecca Riot , because I've never heard it in Essex) but seems to suggest they might in North Wales at least. I'm also intrigued by the idea that they do down in Margate and Herne Bay. Probably not a good enough reason for a day out there though.
Anyway, Priti Patel, what a cunt eh?