It should do, but occasionally doesn't.
If there's an STC from an overseas authority that CAA considers competent (which does include the US and, against all odds, the French) then they should only be doing a check on the basis of approval and so long as there's nothing odd about it approval should be pretty much automatic once you've translated the paperwork into English for them.
They'll still charge the £400 fee, and occasionally some pillock overseas decides that they won't trust the CAA with copies of their approval reports to check (which I think was the reason for the much publicised German silencer business) - in which case yes, you're back to square one. Always worth asking that question before parting with any cash!
G