Try the man with the stall in Hays Galleria. He’s the only local one left. Weekdays you could try looking about Hatton Garden, but they are nearly all closed on werkends Other than that, try Austin Kaye on The Strand.
Try Timpson at the side of the entrance of Tesco's on the Old Kent Road.
Since the closure of SDV Jewellers on the Lower Marsh I now get my watch batteries fitted there . I have taken the more expensive guarantee for life deal and on the ticket it quotes some of the repairs they undertake.
Just to let you know that I went to Hay's Galleria and found out that the stall is called Harry's Watches ( there is a website (harryswatches.co.uk). What a find (thank you boroughbloke for the recommendation).
He was both friendly and knowledgeable. To cut a long story short he managed to repair my watch. I had previously spoken to the manufacturers and was advised by them that they could not repair it.