I wouldn't know my crochet from my musical crotchets but have you checked out the (UK-based) Crochet Society's website? (Despite the word 'society' in that name, it's actually a commercial venture but it appears to offer free crochet patterns anyway, including some for beanie hats). I've no idea whether the patterns they provide actually originate in the UK or the USA but they might well be worth a look:
https://crochetsociety.co.uk/category/free-crochet-patterns/
(I've checked out their registration form. You don't have to sign up to any of their paid services to access the patterns).