The BEF actually had a HEAT rifle grenade too, didn't they? It was (foolishly) directed to be fired from above, like mortar, however. The same tech combined with rocket propulsion could be... tasty.
The No. 68 grenade which had a 2.5-inch diameter and weighed 2 pounds... nope, don't know what you're talking about.

Actually, it didn't enter service until November 1940 anyway.
The British army used the 2" mortar and replaced it with the 51mm mortar, they then decided to phase it out and replace it with 40mm grenade launchers. However they found that while the GL was a useful tool, it simply wasn't a full replacement which is why they purchased US 60mm mortars to give the infantry an organic indirect fire weapon. Much better to bite the bullet and either reduce the number of riflemen in a platoon or increase the size to add a mortar team.
In OTL, the 2" mortar entered service in 1937 (assuming we neglect the Mark I that came in in 1918, and went straight back out in 1919) as a way to give platoons more firepower than a rifle grenade.
TTL, the Dunlop was created instead; the indirect fire gap still exists, as does the ability to reach beyond 200 yards (the 2-inch was good to 500 or so). Since the 2-inch mortar was never in service, nobody feels the lack of it just yet. Ideally, the platoon would have a mortar team and a rocket team, yes - but the prewar funding environment made it a case of one or the other. By the end of the war, the British Army had both... and whilst the PIAT as we know it is going to be stillborn, its' story isn't irrelevant.
