Oh I'm sure the Cortébert watch is perfectly genuine - the face, dial, hands, case, movement, etc. - are all of a piece. But - and here's the point - I'm not convinced that the watch was purchased by the British Government as part of the consignments of Swiss (and American) watches which they bought for the troops. It's in the style, to be sure, but I've also seen lots of "ATP" case backs, and the engraving on that one looks too crude and too new - to me - to be genuine. A watch without the military markings might be worth, say, £85 or so - but around £150 with the markings - so there is a real markup to be made with the extra "ATP"!
Note that I'm not saying that the eBay vendor is a faker - not at all - just that I've never seen such crude engravings on either UK or US military timepieces. He may have acquired it in good faith, but - to me - it doesn't look right. After all, anyone could do that and claim it was genuine.
Here's a specimen of the real thing on eBay - note the clarity of the engraving:
http://www.ebay.co.uk/itm/A-WW2-UNITAS-BRITISH-A-T-P-MILITARY-WRISTWATCH-c-1939-45-NEEDS-A-SERVICE-/201346761922?pt=LH_DefaultDomain_3&hash=item2ee133c4c2
Will