Is this expression generally known and used in Britain?

Well the generic expression is 'you can't swing a dead cat in here'...but you can make it specific to a context by using that structure there.
For example, I just saw somebody on wordreference say 'You can't swing a dead cat in Hollywood without hitting a celebrity' which I found quite funny. So yeah...it's a well-known expression but not very commonly used, especially by the younger generations.

The expression is, there's not enough room to swing a cat. The thought being the 'cat' in question was a cat of nine tails (not much comfort to the person being whipped) rather than an unfortunate moggy.

No, but it presumably originated from a misunderstanding of the British expression "there's no room to swing a cat in", referring to the naval punishment of whipping a sailor with a cat of nine tails, a whip with nine knotted lashes, first recorded in 1695

No - the English expression is - "There isn't enough room in here to swing a dead cat."