Here is the only part I know:
In London, In London
The King is still in London town
Until London bridge comes falling down
The song was written in honour of King George VI and Queen Elizabeth who decided to stay in London at Buckingham Palace during the Blitz rather than relocate to a safer location.
I have tried looking everywhere for the lyrics but without success. The above part of the song is often used in episodes of "Dad's Army"
Perhaps some of the older members like Brian Ross or Edward N. Kelly might be able to help?
Best regards,
David