The registration number belonged to the chassis. IIRC so did the fleet number, there was a small plate with the body number somewhere near the engine.
It was actually a bit more complicated than that when London Transport was overhauling RT, RF and RM family buses.
The same combination of registration, fleet and chassis number would always* stay together, but that identity would go in at one end of Aldenham and a newly overhauled bus would go out the other end pretty much as soon as the tax disc could be got to it. (At that time, the registration number plates on LT buses consisted of white letter / number transfers applied to a blank plate, so registration number plates were effectively made as part of the overhaul process and did not need to be moved from one bus to another. This did however lead to a few clangers that got out on to the streets before anyone noticed - most famously
this one)
Some bus identities were out of circulation as 'works float' for years at a time to cover the legal identity of the kits of parts that were in Aldenham at any time.
The visible brass plate carried the chassis number that matched the log book for that registration number, but these also either got moved from one chassis to another, or new ones were made to match the registration number of the newly overhauled bus. There was a 'secret' chassis unit number on another brass plate, so LT knew what was what.
The body number was applied (to RT and RM group buses) in white transfers, on the inside of the canopy (the driver would be able to see it if s/he looked left, if that makes sense) - RFs must have carried it somewhere but I can't remember where.
While I don't think anyone has ever produced evidence that LT had legal authority to do these identity swaps, but the (then) Ministry must have known about it, and in the early 70s when Aldenham was overhauling buses for London Country, they didn't do identity swaps with them.
The only exception (other than possible flukes) was that buses that had a 'GB plate' having done an overseas tour usually got to retain their original chassis / body / identity -
RT 1702 and (I think) one other survive in preservation.
Quite a bit more about it on the
Red RF website (written more about RFs but relevant to RT and RM types as well.)
* - before anyone else says it, there were some RFs that got their fleet numbers changed around when converted from bus to Green Line or vice versa, or converted for OMO with the idea that all Green Line RFs ought to be in one block of numbers, or something like that.