In 2003 I worked for the RAF and MOD (apparently two organisations known to each other) with an overlap of seven months. Payment from both bodies being totally normal.
As others have pointed out, I got permission from the Staish (in the form of a memo from OC Admin - a very very nice lady!) for the QR that allows you to work during your leave if it does not affect your service role.
With the help of WO PSF (about the most professional bloke I ever met in the RAF) I rolled up every bit of leave and resettlement I could find and with the help of my Sqn Boss got to take it all without a gap.
The only obstruction was the Station Resettlement Officer who said I was breaking in the rules. He was wrong and I worked round him.
Last Day at Work for the RAF 15 Nov 2002
First day paid by MOD 4 Nov 2002
Last day paid by RAF 13 May 2003.
Kerching
None of this would have been achieveable without a great deal of help and assistance from people in key positions, so the moral of the story is pretty clear.
PS The taxman does not give a toss where you work as long as you fill in your tax return correctly.