Part of the problem is that the British Army's main role was Colonial policing not preparing to fight in Europe, and the Cavalry Regiments were regularly rotated from home service to colonial service, and their role in the colonies was mainly mounted (on horses). So units that had just converted to armoured were reconverted back to horses while the units that they were replacing would then come home and convert to armoured. This meant that new recruits had to be able to ride, and were trained as Calvary before training as Armoured. When the war came a lot of the Cavalry Officers were found to be unable to convert to Armoured and had to be replaced, they just weren't interested in vehicles, and couldn't do their job.
I think it would have been a good idea to give Cavalry Regiments Reconnaissance, Screening, and Flank Security roles, they seemed to do well in the Reconnaissance, Screening roles with Armoured Cars, and I think if they had mixed Regiments of Tanks and Mounted Rifles they would have done well in the Screening, Flank Security roles. I'm thinking of a Regiment with two Squadrons of Tanks, and Two Squadrons of Armoured Rifles. The Armoured Rifles would need to keep up with the tanks cross country, and have armoured protection from direct fire. The Dragon Carriers as used by the British Artillery between the wars, and early war would be a good vehicle for this role suitably modified.