I suspect this may be part of the case; though doesn't the UK have some master list of companies? Maybe that could be searched.
In my local experience in the US Barnes and Noble and Borders killed all the non-second-hand bookstores, and then Amazon mostly killed those, but some of the second hand bookstores have survived.
Also, do you count Goodwill as a "second hand bookstore"? They have usually two or three aisles of books, after all.
It returned 3,437 results for me, but this will include inactive companies and chains, and exclude unincorporated shops. Tax records might be more thorough, if there are a lot of unincorporated companies or sole traders that file tax returns, but I don't think those records are public.
In my local experience in the US Barnes and Noble and Borders killed all the non-second-hand bookstores, and then Amazon mostly killed those, but some of the second hand bookstores have survived.
Also, do you count Goodwill as a "second hand bookstore"? They have usually two or three aisles of books, after all.