That's why you list references from your last job and people call them to see what kind of person you were while there.
You don't want your future company to call your current company to give them a heads up that you are leaving.
For this very same reason, companies never ask for reference from your current job.
The UK has a custom to perform an employment and criminal check by a 3rd party company when you join a new company (after you signed a contract and already gave your notice). That's unrelated.
Sure they do. They just give you an offer first and make it conditional on a reference check.