There seems to have been several pongo units with forage caps, when I've been searching, including the Pioneer Corps, and in photos of the BEF.
I suspect it is Army rather than the jolly what oh, boys.
http://www.astonbrook-through-astonmanor.co.uk/pioneers_1.html
There's a bit of info about the British Expeditionary Forces in France early 1940 here. This includes detail of some Territorial units arriving.
https://www.britishmilitaryhistory.co.uk/docs-france-norway-1940-british-expeditionary-force/
The photo may have been taken in France, as per the postcard. The British were in France until May 1940.
War records can be researched here.
https://www.forces-war-records.co.uk/search
My immediate thought was Royal Pioneer Corps, which became Logistics Regiment. I've had a quick look at images on Google & the forage caps look similar. It's worth a look, anyway.
The dates would fit.