We tend to use the same phrasal verbs in more or less the same way, with a few variations. Here are a couple of differences:
'Round' and 'around' have the same meaning, but AE uses 'around' while GBE often uses 'round' as an alternative. So an AE speaker might say 'get around to doing something', while a GBE speaker would say 'get round to doing'.
GBE has the phrasal verb 'get on with someone' - meaning to feel at ease with a person/have a good relationship - while AE has 'get along with someone'.
As a rule, speakers of GBE will be familiar with AE expressions via the media, and younger people may use them. The reverse is not generally true - people from the US and Canada will tell you that English expressions used outside the North America sound 'weird'. :)