As
@Roger1973 says . . . Clocks Back (or FallBack as I used to refer to it) means that, at 0200, it will suddenly become 0100. Effectively, any day buses still running will simply run their mileage out.
In London . . . any night buses duties will extend by one hour to avoid a gap between last night bus and first day bus. If this will result in a duty breaching drivers hours or their "agreement", then a new driver will be required. In practice, a few extra duties will step into the nights' "cut" and either extend duties, or give a mealbreak to the original driver so they can resume and finish off one hour later. The drivers will be instructed NOT to reset their watch, and to ignore the time on their phone or the ETM. In practice, night bus drivers are very clued-up about what is necessary, and need little extra supervision.
In the spring, we have Clocks Forward (SpringForward), which is much easier . . . at 0200 it will suddenly become 0300! Because early day buses will start at their normal times, there is an overlap between nighters finishing and day buses starting, so the relevant night bus duties will finish one hour earlier to avoid duplication. There is always some duplication / overlap, but keeping it to a minimum is the aim. Where a night bus driver works an early trip on a day route, shen some "tweaking" will be required, but this is usually pretty minor.
What about pay? Ah . . . in order to get the co-operation of the troops . . . any driver finishing later than their expected time will be paid overtime. Any driver finishing earlier than their expected time will NOT lose any pay! This might seem weighted towards the driver, but the amounts of money involved are tiny, so it really doesn't matter.
As far as Countdown and Real Time information is concerned . . . when I last dealt with clock change schedules (in Autumn 2019), it was usually left "in the air". We didn't usually update the schedules that were uploaded, but issued manually-adjusted dutycards and schedules . . . although I believe that, since then, it has been expected to upload corrected schedules. Quite how BusTimes deals with buses that effectively jump forward one hour, or stand still for one hour, I really don't know . . . and as I'm tucked up in bed, I really don't care!!