Goodness me, this is quite the topic to tackle, and I'm afraid rather beyond my abilities to cast my net both widely and deeply. I can however, offer some information regarding my own research niches, namely British armoured doctrine, and also the North African Campaign, and perhaps some of the other historians on here may offer their insight into other topics.

So, to begin, we must first make a cursory survey of British armoured doctrine at the outbreak of the war, its development through the interwar itself an interesting and complex topic. The role of the tank in British doctrine at the start of the war was largely envisioned as a 'cavalry' one; the provision of reconnaissance and screening elements for their parent divisions, as well as serving as flank support, covering the vulnerable wings of the main body of the army, and harassing enemy formations to prevent their effective deployment. This was largely a product of wider British doctrine, which established the infantry as the central axis of the army around which all other units pivoted, the influence of the so-called 'Tank Corps Radicals', who argued for the primacy and power of the tank as an independent weapon, and the restrictive and rigid structure of the interwar army, which had seen the cavalry converted to tanks rather than the Tank Corps expanded. This had a number of interesting consequences for the structure of British armour, the most prominent of these being a bifurcation in British tank design - due to the demands for both a mobile, wide-ranging cavalry-style tank, and an infantry support weapon similar to the first World War, British tank forces were hence largely split into army tank battalions and divisional cavalry, the former for infantry support, the latter for recon and flank cover. Thus, Britain's tank forces were relatively unconcentrated, usually referred to as being 'penny-packeted' among the rest of the army, with the exception of the 1st and 7th Armoured Divisions, the two great interwar experiments with concentrated armoured forces.

So from this start, how did Britain innovate with regards to armour? To begin with, very little; the 1940 campaign in France and the Low Countries, though disastrously highlighting the shortcomings of much of Britain's armour, which was undergunned, under-armoured, outdated, and mishandled, was largely overshadowed by the wider disaster of the campaign in general. What changes to the organisation of armour were suggested were rejected, in part, French suggests, because they were seen to be copying the Germans. Moreover, this failure to adapt doctrine was accompanied by a more serious crisis of tank production, as the BEF had brought only 13 tanks home with them from France, of an original number of over 700. In order to combat these desperate shortages, British tank production continued to churn out existing models of tank, many of them already outdated, setting back the already sluggish development of more modern tanks.

If the French campaign failed to overturn the pseudo-cavalry doctrine of British armour, the early phases of the North African campaign was in some ways to entrench it. 7th Armoured Division, as part of General Richard O'Connor's Western Desert Force, inflicted a series of punishing defeats on the Italians from December 1940, manoeuvring in precisely the sort of cavalry strike role that British doctrine envisioned. This was not to last, as where these tactics had worked against Graziani's poorly-equipped, unmotivated Italian 10th Army, they foundered against the newly arrived Afrika Korps in 1941. Outmatched in armour, and with Rommel utilising a mixture of tanks and anti-guns in his formations, British armour failed to find success.

The good news about lack of British success with tanks against Rommel was that this prompted change. Unfortunately, the British initially perceived the problem as the solution; the mobility and striking power of German tank units encouraged 8th Army (the former western desert force) to try similar tactics, creating the 'Jock columns', small brigade groups of units intended to strike from unexpected points and retreat. Unfortunately, what the British had failed to grasp was the need for combined arms coordination between these small groups, and consequently, they were hugely ineffective.

Nevertheless, following Operation Crusader in Nov.41, the British slowly reformed their armoured doctrine. Manoeuvre was emphasised less in a tactical sense, firing on the move largely abandoned, and a significant emphasis was placed on use of terrain, force concentration and firepower, the latter in particular being aided by significant improvements in allied tank design, notably the Cruiser Mk.VI Crusader and the M3 and M4 medium tanks. Additional emphasis was placed on all-arms cooperation - tanks, infantry, artillery, and other branches could and would cooperate with each other, a notion underwritten by the addition of integral motorised infantry and artillery to armoured divisions, thus re-balancing them away from the 'all-tank' concept championed by the RTC in the 1930s.

Refinements to British armoured doctrine paid dividends, even during the disastrous Battle of Gazala in May 1942. The 10th Hussars, partnered with 11th RHA, smashed through a line of German anti-tank guns on the 28th, aided in no small part by the smoke screen and suppression fire of the horse artillery.

The steady improvement of the performance of British tank units was further spurred by the arrival of Bernard Montgomery in August 1942, not from radical reform, but by insistence on training, regimes of which had largely been neglected as most units were desperately needed on the frontline. Monty summarised it thus: ‘although the Eighth Army was composed of magnificent material it was untrained; it had done much fighting, but little training’. Monty also was a driving force behind the eventual reunification of the 'cruiser' and 'infantry' tank concepts with his suggestion of a 'universal tank' - in effect, an MBT, which was largely reflected in his willingness to adopt the Sherman, and later Cromwell, as both infantry support tanks and mobile weapons in their own right.

Although I somewhat oversimplifying, the pattern of an increasing emphasis on cooperation and flexibility in British armour can largely be seen for the remainder of the conflict. Tunisia, Italy and Normandy all placed a far higher premium on infantry/armour/artillery cooperation than before, and this was reflected in the frequency with which infantry units were 'paired' with an armoured regiment for mutual assistance. Thus, despite misfires, such as Operation Goodwood, or the assault on Coriano, and the almost mythological perception of German technical superiority in tanks, British armoured doctrine evolved consistently throughout the war into a functional, if not perfect, series of operational concepts, and bound together by a culture of flexibility, largely enabled by the loose centralisation of British regiments at that point, which enabled innovation and adaptation to meet changing circumstances.

I hope this short summation of some of my own work has been of interest to you, if a tad brief (it is in part a very foreshortened rundown of one of my theses), and I'm more than happy to point you in the direction of interesting literature on the topic if you would like.