With a 1936 starting date, funding the RAE and other organisations to get pressure carbs or injection for aircraft engines some time in WW2 is viable. I suspect something could also be done for turbochargers and turbine technology in general but superchargers will probably remain massively important for much of the war.
For ground equipment, I'd argue that funding new/expanded production facilities, expanding the skillbase of institutions (Mechanization and Artillery) and getting more manpower and equipment is the most impactful change one can make. Officials like Chamberlain did start rearmament early (1934 or even earlier) but deliberately prioritized the RAF and to a lesser extent the RN, at the expense of the ground Army. There was a decent rationale for it but even this ended up not being enough for the RAF or combined Anglo-French airforces to oumatch the Luftwaffe as one may have expected from this prioritization (notwithstanding the fact the UK still made an extremely impressive air effort).
However, the Limited Liability policy of 1937 which refocused Army buildup to the ME/Africa hurt efforts towards a new BEF, some measures were not taken until the occupation of the rest of Czechoslovakia in Spring 1939 (expansion of territorials). It had been assumed that France (and Eastern European countries before Munich and the invasion of Poland) could deal with ground forces, but the reality is that France alone couldn't be expected to do it alone on the basis of its smaller population and industry compared to Germany. The wartime 1939-40 plans to grow the BEF to 20 and then 36 divisions after the first and second years of war (with plans for 55 divisions by year 2 not applied in industrial terms) were late and meant the ground balance would remain concerning until 1942 at best.
Timing is also of the essence for the year of maximum danger (1940). IMO, Entente forces OTL were such that just a few extra British divisions/an extra Army by Spring 1940 would reach the threshold for France to reinforce weak sectors or create an additional local reserve (as all mission requirements should now be met).
The part about expanding the skillbase in institutions is crucial. Development of the 6 pounder gun could not go as quickly as it could for lack of sufficent personnel, the limited ressources being prioritized for artillery and AA. Vulcan Foundry had only 2 draughtsmen to work on the A12 infantry tank for 6 months. The Ministry and Supply and other organisations clearly lacked skilled personnel for a while in the early war and prewar periods.
Today I was reading a British wartime assessment of tank development found in the national archives by Ed Webster. This problem of a lacking prewar tank development structure was well identified. Funding wasn't even enough to build one prototype a year during the 30s.
This can be contrasted with the French who, for all their ill-conceived ideas, managed to build up a strong technical research and industrial base for tank and armament developments from 1935 onwards with the Armament Direction (DFA), state design bureaus (APX, ARL, AMX, Bourges) with good partnerships with private industries (ARL partnering with Batignolles-Chatillon and Fives-Lille for turrets), and an industrial base which was already getting plenty of building experience from 1936 onwards. One can only wish the British had such a base to put their more open-minded ideas to use early on.
In analogy to French efforts and post-1939 British reflexions, one could obtain:
- a standardisation of turret designs for a given period, armament and turret ring size. All British tanks until 1941 bar Valentine had a near-54-inch turret ring diameter and a 2 pounder+coax armament with 3 men, but no real standardized turret (which was regretted later on). Ideally one could have one or two generations of 2 pounder turrets, and then a 6 pounder turret, ideally anticipated with a minimum 60-inch turret ring diameter to house 3 men; with the armour being adapted to the recipient tank (as with French APX-1 turret family with 40, 60 and 100mm bases, ARL-2 with 40, 60 and 75mm bases).
With the ressources needed for an early 6 pounder (if it doesn't outright start in 1936 as a closer contemporary development to French HV 47mm), development of a suitable turret and tanks could proceed before the war, possibly as early as the 1937 French studies for HV 47 and 75mm turrets.
- Early and timely deployment of armor piercing and ballistic caps for AP ammunition to anticipate thick, sometimes cemented armor, as well as HE ammunition in spite of their weakness in 40 and 57mm calibers.
- Suitable specifications and eventually deployment for thick cast and welded armor assemblies (French spec for 40 and 60mm castings in 1937, first welding spec in 1938). The British as is already had know-how for both prewar.
- a proper review of rail transport capabilities (the restrictions used for the first half of the war were based on outdated, overly restrictive assessments)
With the 25-ish tonne bridge and weight limit for A12, the British had more leeway at the lower weights than the French, and probably a higher weight category could be identified (their transport categories were at 20 tonnes and 35 tonnes which was updated to 45 tonnes). But this doesn't do much good when the limit isn't used by the Cruisers (Covenanter and Crusader designed to a 18 ton limit), or misused by other designs (A14 and A16). The A9/10 and A13 are screwed by their origins, but the new specs definitely need to use the 25 ton limit.
The 1937 proposal for a 40mm minimum armor thickness requirement should go ahead ASAP (the maximum amount of combat tanks should have more than the 14mm or 30mm bases), even with the historical recommendation of 70mm which had been used as a reason to not demand 40mm in 1937 (note: close match to French 40 and 60mm bases for most late 30's projects).
The A9s and A10s are good insofar as they can be produced in numbers in time for 1939/40 before newer designs.
Vickers could possibly have doubled down on the initial idea of the A10 as a light infantry tank, especially if the armor basis is reinforced to 40mm or as much as a 16 tonne weight limit could allow (functionally earlier 3-man, less armored Valentine), on the grounds of the suspension and engine not being that conducive to high speeds and the need for a lot of decently well armored tanks. As infantry tanks of the time didn't require a biw LG, the more efficient sloped front armor of the A10 prototype could be retained unless a full-width driver's visor plate is still needed (as A17 Tetrarch prototype to production). Val could still go ahead later. 1936 might not be too late to get A9 and A10 with diesel engines as originally intended by Vickers and finally done on Valentine. The AEC petrol engine used in the former was actually derived from the diesel of the latter, but not much power was gained and I'd argue the lower fuel consumption and improved characteristics of the diesel version would actually be preferable. If petrol is retained, the RR Phantom III (II was used on prototypes) would be more efficient than the AEC petrol conversion.
At one point an A9 without the MG turrets had been mooted as a cheaper version, and an A15 Crusader with an A10-style MG mount was mooted before and after production started. Killing the MG turrets before the war was probably doable.
For engines, the example of the V12s based on petrol truck-engine blocks (Bedford and Meadows flat 12s) could have been applied to diesels (Matilda IIs for nearly 200hp or AECs for 260 hp before the midwar upgrades) to at least get a more powerful unit and a more space/weight-efficient setup than the twin engines on Matilda II.
But the UK will have to spend effort on a powerful engine regardless, be it special or adapted from aircrafts or railcars.
Starting the purpose-designed Leyland, Vauxhall, Harland & Wolff 300-350 hp diesels from 1940-42 earlier could at least have improved the early war designs (A12, 25 ton class Cruiser, follow-on infantry tank). Pursuing Harry Ricardo's schemes for dedicated 350-580 hp diesels from 39-41 is also doable. Putting the foot down for a tank Kestrel/Merlin that the Air Ministry would not be allowed to touch works.
Ricardo argued that even the Meteor work may not have been much faster than a dedicated petrol engine, for what it's worth. As Ricardo deplored, the lack of a dedicated tank engine effort also directly led to the plethora of engines and tank designs in use until 1942.
The TL

R of this is that even if the Infantry/Cruiser tank dichotomy can still be relevant, at least having a 25-ton basis Cruiser with a modern engine, 50 or 60mm of armour in places and a 3-man 6 pounder turret in the second half of 1940 as the A13 follow-on is absolutely credible.