I agree with CKLD, a DEXA scan should be available on the NHS if there is a concern about your bone density. I have had several done over the years, private and NHS, always at the same hospital on the same equipment. Currently on a programme of 2-yearly DEXA scans.
If you want to go down the private route... I don't know whether many private hospitals have DEXA scanners but they should be able to quote for a private scan and relevant reporting as they may have a contract to have these done privately at the local NHS hospital. It's easy to ring around private providers and get quotes - I did this for some off-label cancer treatment that I was considering a couple of years ago.