In N-SSA competition, with target benches, loading stands, and your gear all at hand, they give you 5 minutes to load. this includes smearing lube over the bullets in the chambers, but does not include capping. After everyone is loaded, you have 2 minutes to cap. While paper cartridges are allowed, few use them currently.
As has already been said, cavalrymen generally carried multiple pistols and would shoot one dry and then pull another. Even with paper cartridges, I can't imagine trying to load a revolver on a moving horse, and you'd be a target standing still. I doubt these guns were ever loaded under fire except in the most dire of circumstances.
I suspect that even officers on the ground seldom used their revolvers. They were leaders, not fighters, unless things had gone horribly wrong and they were caught up in the actual fighting. But while they had a holster to carry a revolver, they had no accoutrements for carrying extra caps and ammo. It's possible they had an extra box of ammo tucked in a coat pocket. They were not set up to be a continuous combatant with their sidearms.
I don't think cylinder swapping was ever a common thing. I do recall someone finally produced some historical evidence for them being issued or used (I can't remember), but that was extremely rare documentation. There just aren't tons of quartermaster reports documenting the issuing of pistols with matching spare cylinders as there are for the standard issues.
I also don't know how common capping tools were or if they were issued or even existed in period.