The British ‘have‘ a bath, while the Americans ‘take‘ one.

Of course, Americans frequently use delexical ‘have.’ Note ‘have a party,’ ‘have a meeting,’ ‘have a soiree,’ ‘have a rendevous,’ ‘have a shindig.’ For reasons that are unclear, many of these seem related to socializing. So Americans say ‘have a shower’ if they are talking about a wedding shower, but (usually) don’t say ‘have a shower’ when referring to getting clean. Not sure if this constitutes a pattern, but it’s a striking trend.

На деле же оказывается, что подобные фразы отличаются по смыслу (подробнее здесь ), например:

‘have a drink’ and ‘take a drink’ (the latter usually in the context of ordering said drink)
‘take lunch’ (meaning ‘taking a break for lunch’) and ‘have lunch’ (meaning the act of eating lunch).