I'm standing by my non issue, custom made cigarette wallet. Tobaco doesn't need the frame to keep it from being crushed.
Caryl, as you can imagine I have had very many suggestions about the pouch. If you saw it I'm sure you would accept it was not for cigarettes, it is much too narrow and even after almost 70 years I would have expected an aroma. It is just less than 1/4" thick.
You can discern a slight indentation on the back approx 2 1/4" in diameter, as if it once contained something circular.
I am awaiting a reply from the Imperial War Museum here in the UK, having asked them if it is recognised by anyone there.
Colin.
AH, I was not aware how thin it was.
Back to the thinking board. Wallet that held his lucky crown? (money, not headpiece)
COMPASS CASE !
OH OH OH Pocket watch - family piece, he'd want it protected! He would have removed it when he got home. While not made for a watch, it would have worked!!
MYSTERY SOLVED!!
Email to Alaskan Gal:
Nancy,
I have just returned from an 80 mile round trip to Cheltenham and the BBC Antiques Roadshow.
It was worth every mile as I have solved the mystery of the pouch. It contained a mirror for the use by a lady in her handbag. It was known as a "sweetheart souvenir" and the non-official RAF Crest on it was used at the time of WWII on various items like makeup compacts, lipsticks, brooches etc. sold by commercial enterprises.
I am so pleased to be able to let you know what the Militaria expert said - so now we know. I hope it may give you an idea of why it was in your dad's trunk, do you think so ?
Colin.
Colin, this is so cool. After all this time you were able to discover what this was. Nancy will be so thrilled, as am I.
Bravo, good job mate!!!
Make sure you send a PM to Nancy, so she can see this.
I have sent a PM to Nancy here M1, as well as the email I sent earlier copied above.
I got a nice reply to my email to Nancy (Alaskan Girl) and she is so pleased we have solved the mystery. In fact it fits in with what she knows abour her dads time in England as he almost married an English girl!
Colin.
Ah, "almost married an English girl!" I didn't know that. 
I'm sure you made her very happy, for I know how appreciative Nancy is.