The story about Berkshire losing its royal prefix seems to be apocryphal, albeit quite well-known! No club once given the 'royal' designation (in the British context) has subsequently lost that designation, unless the whole institution has given up (as at R Cornwall and R Isle of Wight). What has been (much) more controversial has been the status of royal clubs in former colonies and/or British overseas territories: Royal Dublin (for example) has adhered to its original title despite the post-1949 status of the Irish Republic.
Royal Troon is the most recent 'royal' creation within the UK, and actually much more recent than many visitors realise (as is Birkdale): when Tom Weiskopf won the Open there in 1973, it was still simply 'Troon Golf Club'.
As for 'regional' (as opposed to county clubs) what is now Lindrick was officially the Sheffield and District GC until the 1920s (and there still used to be teeboxes around with 'S&DGC' visible). Not sure if either East Devon or West Cornwall has yet been cited in this thread context either: the G7 dignitaries will have the enticing opportunity to play the latter (COVID permitting) when the rulers of the world visit that part of Cornwall later this year.