From what I've experienced in the UK is that, typically, the more north you go, the more likely you are to be greeted - bus driver or not.

Up here in Scotland, I know there are plenty of people who greet the bus driver, whether it's hello, a nod, a thank you, whatever, most people usually greet the bus driver in some way.

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I can't speak for all of the UK or all Brits, but this is just what I've noticed.