I wonder why your family doesn't say it if it's the 'British' thing to do?
Well, I guess it's partly a regional/class/accent/generation thing as well - different people have different ways of greeting others, and, as chary said, not everyone uses the same phrases.
I guess my parents just weren't brought up to greet people with 'You alright?' (presumably because it was considered impolite to do so - my mum's family especially were big on etiquette and being polite, due to her grandfather being a bishop and her parents both being ex-military). Instead, they are much more likely to greet people with something more along the lines of: 'Hello, how lovely to see you. How are you?'.
These days, my brothers both say 'you alright?', and I guess I say it sometimes, but as kids it wasn't something we really heard or encountered... we just picked it up as we got older and met other people who used it.