I would be very careful going down the ‘their culture is similar’ route. Even if this was true, which it almost certainly wasn’t, a whole host of other factors are going to make such comparisons meaningless & have frequently led people down wrong paths.
The Tatars were a power at this time, heirs to an even greater state, and the close ally of a ‘superpower’. They were a relatively rich, organised large state and had easy access to firearms, gunpowder, etc. The Bashkirs and Comanches were isolated relatively poor, relatively small groups with little resources and no significant allies. I know what I would look at to explain why the Bashkirs and Comanches (assuming they did) stuck with bows.
I think the Tatars continued to train as mounted skirmishers right the way through but to put a date on when they switched from bows is more difficult. Going by impressions only the light horse of this part of the world started to get reasonable numbers of firearms by circa 1600 and switched by circa 1650. But of course with many exceptions.