TBH I'm not really buying the whole 'HESH is shit' line. If HESH was so bad and so important than Britain would replace it with something better between 1965 and 1990.
My guess is that HESH served Britain's needs well enough that it wasn't worth the effort and cost of changing it when there were other things to do; like the DU KE round that was issued at 12 per tank in 1991 PGW to take on the T72M.
EDIT: Just to clarify, it isn't a simple matter to swap ammo types. Britain would have a 30 day (or whatever) war stock of HESH laid in and modest annual production would cover practice use and 'shelf life expired' replacement every year. If Britain decided to go to HEAT they'd have to undertake a huge initial production run to lay in substantial stocks of HEAT ammo to build up a reasonable war stock and cover the conversion training and large follow on order to phase out HESH at the same rate and Armoured Corps personnel are inducted so they don't have to train new gunners on HEAT, HESH and KE.
I was trawling through the world wide web basically looking for 'why HESH?' but found very little and there was no angst about it on ARRSE that I could find - which generally means there is nothing wrong with it.
I watched several videos where they were shooting the stuff out in Iraq to use up ammo reaching end of life and in one the Rupert went on about how great a round it is - I do not know if he was spouting the party line or what?
But it does seem to be the general refrain - 'HESH is great'
So I am not convinced thus far that HESH is a bad choice and the British Army do seem to rate it despite the rest of the internet telling them that they are wrong!
Okay back to the wider question
As part of my searching through the interwebs using my black belt in google-fu and I read one thing where a former officer in the Army during the 80s spoke about how they were asked to develop the army's requirements for the Chieftain replacement (Challenger II IOTL) as Challenger I had at that time been a bit of a disappointment and they basically told the army 'Buy/build Leo II' but if building a domestic design 'use the same gun as our allies' and of course were subsequently ignored.
So what I would suggest is something along the lines of having ITTL a Vickers led designed Torsion bar/RR V8 'Chad' Chieftain instead of the Leyland led Horstman/L60 Chieftain entering service a few years earlier and being a far more successful tank leads to the SHIR 2 being developed for the Iranians slightly earlier with 1200 units being made (a later or no Iranian revolution?) and the British army are able to leverage this design for TTLs Challenger I making 900 odd by the end of the 80s - Based on the Shir 2 (Lion) the British call this 'Black Prince' with a Vickers derived turret arrangement and sighting system but otherwise similar to the Challenger 1 of OTL
The Shir 2 gains quite a reputation during several boarder conflicts with Iraq during the 80s and Kuwait also buys a number of these tanks to supplement their Chieftain Fleet and these orders are followed by several other middle eastern nations (Jordan and Oman for example) - with a total of 2300 units made of all marks
This tank using the L11A5 gun of the later Chieftain is intended to supplement the existing fleet of tanks and not then replace it.
The Iran Iraq wars flair up throughout the 80s and the Iraqi invasion of Kuwait and Saudi Arabia in 1990/91 sees multiple examples of Lion /Black Prince in the service of the armies of Iran, Kuwait, Oman and the British Army verses Iraqi T55/T62 and T72s with the Lion /Black Prince almost always prevailing in the majority of engagements.
With the fall of the Berlin wall and the reduction of the BAOR and downsizing of the British army all Chieftain tanks are withdrawn from service by 1994 with the final unit based in Berlin ending service as that garrison is finally withdrawn.
In the Mid 80s the decision is made to align the future British tanks with that of the rest of NATO and build the Leopard II an under licence version of the then latest Leo II using the longer Rhm 120mm Smoothbore L55 gun and this tank goes into production (despite the wailing of the Daily Moral panic) during the late 90s and 420 entering service during the early 2000s as effectively Leo2A6's
Subsequent events see the Black Prince (Challenger I) completely withdrawn from service during the noughties as the British army radically downsizes and many of these are sold off to armies around the world
At time of writing the British army maintains about 250 modernised Leo IIs in 4 Regiments (Battalions for those of you who are hard of Commonwealth) - most of them to an A7 standard with 55 of them with 'street fighter' urban warfare modifications for use in the Iraq occupation force and a number of support tanks using the same hull with the remaining tanks kept in a pool of un modernised tanks.