Miles away from the UK, here, but this sort of situation is guarded against in our neck of the woods. Where the weather is suitable for a visual circuit, the overshooting aircraft does that, and the tower controller provides visual separation in the usual way. Where circling minima doesn't prevail a departure will not be cleared to go once an arrival is inside a pre-set distance, which varies depending on the types and runway. This can be fairly conservative, as terrain nearby limits early variations from the standard miss. The distances have been determined by placing multiple scenarios on the sim, then rounding up the distance required, to pretty much guarantee that the radar separation will be maintained throughout.
The downside is that when the weather is bad the delays can start to mount up, as gaps have to be created between arrivals to allow for departing flights.
It's a bit of a blunt tool, designed to guard against a situation that doesn't actually happen that often, but the view (correct, IMO) is that if the weather is below circling, an aircraft can make a missed approach at pretty much any time, for any number of reasons, some of which mentioned above, and when that happens, even if it's just once a month, it should not be necessary to treat it as an emergency situation.
It's a right pita when the weather is bad and traffic heavy, but it's default-safe.