BBC London had a report on how a BBC engineer, the late David Martin, edited King George VI's recordings, to help cover up his stammer.  Watch the report here

Is there anyone around with the knowledge and kit to demonstrate the skills needed to edit the material?

I know there is an ex-member of staff who restores old recordings (in a private capacity) for the BBC and other organisations.

The shots of tape machines about 30" into the BBC London piece, and again near the end, show not a tape machine but a Philips-Miller film recorder. These machines were certainly still in use during the war years so could possibly have been used. No idea how easy editing was, but probably somewhat easier than using disks if lots of edits were required. There's an article about them on my site http://www.roger.beckwith.btinternet.co.uk/bh/tapes/pm.htm. Also worth noting that according to http://www.royal.gov.uk/imagesandbroadcasts/thequeenschristmasbroadcasts/ahistor...
George VI did all but the last of his Christmas messages live.

Prompted by Mike Chessher I've found an account of the broadcast in Tom Hickman's book What did you do in the War, Auntie? This adds some extra details to the story. Hickman says that the Christmas message was usually recorded at Windsor (this seems unlikely and contradicts the royal.gov page mentioned above) but in 1941 was done live from Sandringham. Churchill heard the broadcast and asked the BBC to doctor the recording before it was used in the Empire Service.

The task was given to David Martin. He had two versions to work with - a recording of the live broadcast and an earlier take which was to be used if the line from Sandringham had failed. Each recording was spread across four 12" disks; only the outer 2" were recorded on in order to maintain the best possible sound quality.

Martin set up his disks on a bank of eight turntables and would have faded between the two versions as required. He would also have needed to do some jump-cuts, where the pickup is lifted from the playing disk and popped back down a few 'grooves' further on, thus removing a pause or stumble. The 'in' points were marked on the disk with a chinagraph pencil.

This performance was repeated several times - for Africa, India, twice for North America and finally the Pacific which finished at 0620, by which time Martin was understandably 'knackered'!