(I should specify the Ordnance QF 2-Pounder.) I imagine the Pom-Pom had a HE Shell.
It's not like a 40mm gun like the 2 Pounder can't have a HE Shell (see the 40mm Bofors.) Now, I understand that Pre-War the Royal Artillery was full-on Gollum about anything High Explosive, but the War lasted six years. Was the gun simply withdrawn from service entirely by the time a round could have been produced?
Initially, the British quite strictly adhered to their tank doctrine. Much like there were infantry and cruiser tanks, there were general purpose tanks armed with 2-pdr cannons and Close Support (CS) tanks armed with 76 mm howitzers for firing smoke and HE shells. For a period of time, this was considered sufficient.
As early as the summer of 1941, however, there were experiments performed with HE ammunition, starting with the larger bore 6-pounder cannon. The effect against lightly fortified targets was certainly greater than AP, but still not enough to be considered viable. One of the issues was that there were two types of fuses trialled, both of which were good against different targets, so two types of HE would have to be carried in order to reach peak effectiveness. However, this was deemed too confusing for the loader, who would have been viable to wrongly select HE when AP was asked for, and thus put the tank and crew in jeopardy.
Perhaps having seen the fact that American tanks do just fine with both HE and AP loaded, trials were performed again in the winter of 1942, this time comparing the effect of 37 mm and 40 mm HE.
40 mm HE fired from the 2-pounder gun gave a rather poor showing. Against a sandbag wall with a brick wall behind it, AP actually did more damage than HE did, as the AP destroyed the sandbag wall and could still punch through the brick wall behind it, whereas the HE shell broke up upon hitting the sandbags and did not damage the brick wall. The effectiveness against other targets was notably lower than that of the 37 mm HE shell. Conclusions of the trials note difficulty in choosing the delay for the fuses, as, like noted with the 6-pounder trials, different delays are good for different types of targets.
Eventually, the British buckled. By 1944 it was clear that even squeezbore 2-pounders were no longer effective as anti-tank weapons, and HE ammunition, however flawed, would increase the effectiveness of old vehicles armed with these cannons. 21st Army Group liaison letter #4 dated January 1945 notes that, once supplies become available, 70% HE and 30% AP ammunition will be issued.
Like you mention, 40 mm Bofors cannons did have an HE round, which directed at least two nations supplied with 2-pounder guns to consider merging the two.
One was the USSR. The Soviet Union received Matilda and Valentine tanks equipped with 2-pounders from the UK. One of the greatest complaints was that these tanks did not come with either APHE or HE ammunition. One of the solutions was to equip them with domestic production 45 and 76 mm cannons, both of which had APHE and HE ammunition. The other was to rig the 2-pounder casing with a 40 mm Bofors shell. Trials were performed, but as far as I can tell, this solution did not pick up any traction. The reason might be the same as why re-armament programs failed: the British started sending CS tanks, along with 76 mm HE ammunition.
The Australians came to the same conclusion as the Soviets did, but actually stuck with it. Bofors shells were fitted into casings from 2-pdr AP shells starting in 1942, and were issued like this.
Sources:
Archive of Canadian Military Headquarters, London : reel T-17473
Archive of Canadian Military Headquarters, London : reel C-5773
Archive of Canadian Military Headquarters, London : reel C-5777
Australian Government Department of Defense DEFENCE UNEXPLODED ORDNANCE INFORMATION SHEET