Another vote for 'it depends' / 'it varies'
Generally speaking, both duty schedules and rotas (these are the most common terms for it on the buses, although there will be exceptions) are at their most efficient when there's more in them, although form a human perspective there can be advantages to smaller rotas.
Smaller depots tend towards having a single rota. larger depots tend towards splitting work in to different rotas. Although the smallest of operations can go to the other extreme where each driver has a regular duty and maybe their own regular bus.
From the London perspective -
I'd agree with what's been said about London - it's broadly traditional for drivers (crews in the past) to stay on one route - as ever, there are exceptions - if some routes have a higher or lower proportion of weekend work (or occasionally spreadover / split duties) then there might be a rota for route X that includes a few duties on route Y, to balance this sort of thing.
In traditional London Transport days, there were quite a lot of cases where one garage only worked a particular route some days of the week, largely to balance weekend work between garages - for example in the 70s, the east London suburban route 86, which served big shopping town centres like Stratford and Romford, needed more buses / crew duties on Saturdays than it did during the week, so Upton Park had an allocation on Saturdays with buses and crews who worked route 15 or 23 during the week - the city commuter traffic wasn't there on Saturdays. Poplar garage in turn had a Saturday only allocation on the 23 because their route 40 had a lower requirement on Saturdays.
Then in the 90s most crew routes were OPO on Sundays, and this work ended up on an OPO route's rota - in turn that route that was OPO all week might have midi buses (especially before Sunday became a busy shopping day), or OPO buses from another garage on Sundays (Merton garage's Sunday allocation on the Putney route 22 lasted a long time after the weekday 22 went OPO and might have been one of the last survivors of this sort of allocation.)
There will be a few routes where either the duty schedule or rota doesn't work efficiently on its own, so there will be joint compilation (duties that can work more than one route) or joint rostering - although this can cause problems for the operator if one of those routes is lost at re-tender, and the drivers on that route aren't eligible for TUPE transfer to new operator because they aren't full time on that route, which may (depending on the current staffing levels) be a bit of an embarrassment.
Some operators have a mix of 'everything' rotas and 'early', 'middle' or 'late' rotas (sometimes it may be one route has multiple rotas, or there may be one 'early' rota for the whole garage) - this is a relatively new development.
Night bus work is usually to be on a separate rota, with some drivers working regular nights.
Even in London Transport days, there were some garages that had an 'all in' rota (obviously crew and OPO work was kept separate) rather than separate rotas for each route, and this had been standard practice in the tram / trolleybus division except at the largest depots where there might have been 2 or 3 rotas, each covering a group of services.
Some garages have a mix of standard and extended hours rotas - on some routes the duties work out best with 9 hour days rather than 8 hour days, so a garage may have a few rotas that offer drivers who want it to work a 45 hour week on a regular basis - or alternatively 4 long days a week, or 9 days a fortnight (although this gets complicated with weekly pay.) This again is a relatively new thing.
Outside London -
There's less of a tendency to stay on just one route (although this can happen with longer rural / interurban services) and there's more likely to be one 'town' rota (or routes grouped, so that there's one rota for the routes on one side of town, or one corridor) and rural routes may form their own rota, or again may be grouped for efficiency. It may be that 'rural' drivers will do a few 'town' journeys to make the duties up to the right amount of time, or to balance out Sunday work.
I'm aware of some operators where expecting drivers to do the same town route on both halves of a duty is not popular, let alone doing it all the time.
Ultimately, there isn't a simple answer, as every operator will be different, and there may be variety between garages of a single operator (particularly where today's operation had historically had been different operators.)
New starter rotas are another relatively recent thing - traditionally the newest drivers went on 'spare' which meant they had to know all the routes and types of vehicle, which could sometimes be a bit much (especially after drivers started as OPO drivers). Then in the late 80s, drivers tended to start on 'mini' and wait for a chance to go on to 'midi' then (eventually, maybe) 'big bus' - each stage coming with a pay enhancement.