Concerning the Pioneer Corps, it was not specifically Jews that were assigned
to it, but generally non-British volunteers as well as regular soldiers
considered unfit for combat (though in fact units did fight alongside the
infantry later) or "undesirables".
Harry Ratner, who served in the Pioneer Corps, says it was virtually a "foreign
legion", or "International Brigade" with even a Spanish Republican company,
Palestinian companies comprising both Jews and Arabs though these were later
segregated, Cypriots, Africans, and some German and Austrian refugees
(otherwise considered "enemy aliens"!)
The officers were all British though.
Harry also mentions that many intellectuals including Arthur Koestler served
with the Pioneers.
("Reluctant Revolutionary", by Harry Ratner, 1994, ISBN 0 9508423 9 7) During
the liberation of Paris Harry, who was of French background, managed to hitch a
ride in with a Free French unit and reach his mother's house, a nice surprise
for her, son Harry arriving in British uniform!
Other Jews served right across the British forces, of course. My Dad was in the
Signal Corps and tended to look down on his brother who served in the Pioneers,
probably through defective eyesight. But this was plainly unfair, especially
as Pioneer units were in the frontline.