Oh, that is a lot of story to tell, a lot going on both in private and work (too much to be able to find any time to be busy with Tolkien related matters). Have been active behind the scenes: reading and following the news, selling and buying books when possible. I'm feeling better, my wife is close to starting to feel better, the company is rolling, house is almost build, books are back here with me, ... well too much for a single post.
An idea... if Allen & Unwin London Office was in Bridgestreet, would this be called Shepherd's Bush by locals? Referring to the district where the signing happened instead of the actual place? If so, perhaps this book was signed at Unwin offices? Tolkien went to visit it in 1962 on several occasions (according to Hammond & Scull).
Television Centre is a building complex in White City, West London that was the headquarters of BBC Television between 1960 and 2013.
White City is a district in the London Borough of Hammersmith and Fulham and forms the northern part of Shepherd's Bush.
Tolkien did his interview in the London BBC TV Studio in the 10 December 1962 visit. I cannot find any evidence at all that there is a second BBC TV Studio other than the main one above, so I would assume this was the visit where he signed the book.
Curious, what evidence/provenance associates the book with Sheperd's Bush 1962?