The UK and the US are not all that different as I read it.
In the US, you can carry out an approach using the aids of a closed airfield without any ATC clearance at any time if flying VFR. Exactly the same as in the UK
In the US, if you are IFR, you need an ATC clearance (from the Tracon or Center if the airfield ATC facility is not available). In the UK, you also need an ATC 'clearance'. The only difference is that the Area Control Centres in the UK can't give you one so there is no possibility of legally making an approach in IFR at an airfield which is closed (except with special procedures agreed with the CAA by individual operators).
Scott or West Coast, correct me if my simplistic view is wrong please