The identity swapping that happened with Routemasters and other type had gone on from London General days, records show that B types had body swaps although generally the chassis kept its identity and registration plate, bodies were freely changed between buses.

However with the advent of Aldenham and the RT family LT used the Works Float system not only to control the numbers and types of vehicles going through the Works but also to facilitate continuous taxation of the vast majority of buses being overhauled by simply changing the chassis numbers fitted to the vehicle on a brass plate to that which was on the log book, in the case of RMs this matched the stock number
When licensed buses arrived at Aldenham, the complete tax disc holder, with insurance and ownership certificate were taken off and fitted to the new version of that identity before it left for its often new home. There are photos of the Licensing Office with tax disc holders on hooks awaiting fitment

The swapping over of tax disc holders on licensed overhauled vehicles couldn't and didn't happen at garages. For that to happen it would have needed every bus to go back to the garage it came from AND the ownership disc and insurance disc to also be sent between garages. They were not! garages were however responsible for fitting new tax discs to their fleet of buses and for informing Chiswick RSE office of long term out of use.
What more likely happened on the Mortlake example was Mortlake were sent the new license for RM 1645 as it wasn't due to go into Aldenham until the 6th. They were told to keep it and when the overhauled RM 1645 was delivered on 3rd March, they put the new tax disc in the bus, RM 1645 being driven down the 45 minute journey on trade plates.

My bus number (if any): RTL 960, RM 1585, RMC 1458 and several RTs