If it was new to London Transport, and that covers everything that was delivered up to 10th January 1995 (the date of the completion of South London’s sale to the Cowie group), it’s a London bus. (The Upton Park SLWs, delivered right across the cusp of their subsidiary’s privatisation, thus find themselves in two camps). Everything after is only half-valid – it’s a ‘bus in London’, in the way that Transport for London only obliquely counts as London Transport.
Found a picture of M 1353 (C353 BUV), that’s been talked about a fair bit; while Hammersmith Bridge was undergoing another protracted period of repair during 2000, a route numbered 509 was commissioned to take people from the station to the north side of the bridge. The fact that the route had to come seven miles from Hounslow just to take people a hundred and fifty yards was spectactularly wasteful, but it was worth photographing, so I have a few shots of the route. This one is at Hammersmith on 29th October 2000.
I remember M 1353 best as a Sidcup crew bus, allocated to convert the 21 from RM in November 1985 alongside the garage’s existing Ts (representing an extremely rare mixed-type operation that became more common as standards slipped). Once Ls came to replace the Ms, it was off to Stamford Brook and settled in what became London United territory, eventually working from Hounslow (as here), Fulwell and Hounslow Heath.