2 pounder HE
was designed, and some tanks in France 40 had them issued. It was judged to be worthless as too small to have any significant effect. Also, the mass panic to replace guns lost in France (that delayed introduction of the 6pr) probably saw an emphasis in producing AP ammo over expensive HE with marginal utility on the battlefield.
(Also - the French battlefield may have been short-ranged enough in general that the co-ax did the job adequately. The desert was different - and that is where the stories of having to charge into closer range start to appear. The desert is also when the "88 myth" rears its head, though mostly the German advantage starts when they use 75mm and 76mm ATG as a matter of course methinks. Most "88s" would be a captured Soviet gun.)
The Australians produced their own 2pr HE of own design later in the war for use on bunkers. It may have been base-fused. However, Australia seemed to prefer to convert as many 2pr Matildas as they could to the 3in close support howitzer version.
2pr HE in UK service was reintroduced later on, though by then the gun was being phased out. It gave armoured cars a little HE capability. Then the Littlejohn adaptor deleted that (unless you wanted to hide behind a house and use a huge spanner on the adaptor for 20-30 minutes!
)
6pr always had an HE shell from the start, but with the primary emphasis on anti-tank it was not a standard issue at least at first. It also had some problems with fusing due to the high velocity (the Russians had a similar problem with their long 57mm HE fuses). Later on, that was sorted. But again the British thought the small shell only marginally useful, preferring a proper 75mm HE to get any significant effect on target.
So HE for both did exist (HE/T Mk II for the 2pr, Shell HE Mk10T for the 6pr), but was simply not issued all that much (same as 2 inch mortar HE ammo). Not seen as worthwhile as too tiddly a round, and the AP ammo was better to have in a gun focussed on AT work in any case. HE rounds were also significantly more expensive to produce, as they needed filling and a relatively costly fuse. (UK AP was almost always solid shot).
The cure for the inadequate HE problem of the 6pr was to
bore it out to 75mm so it could take the US 75mm ammo. The OQF 75mm is a bored-out 6 pounder, is all. HE effect was significantly improved at the loss of some AP effect. But in Europe the bread-and-butter work of tanks was to lay HE onto soft targets and the rarely encountered heavy German tanks could be dealt with by specialist AT units with the 6pr or 17pr.
cheers
Andy
BTW - Tony Williams has a thought-provoking "what if" article on an alternative 1930s British tank gun strategy here
http://www.quarry.nildram.co.uk/alt%...tank%20gun.htm