There shouldn't need to be separate apps. I like Edinburgh, only been there twice and both times as a tourist but I don't think "Wow, a separate Edinburgh bus app" would have been a boon, whereas "Oh, my bus app just works here" would make sense.
At one point Edinburgh's bus operator was part of the same legal entity as the company which provided some bus services in my city, though that is no longer true. London has it right, no tourists and almost no locals care about the bus companies. All the buses are painted the same colour, all of them work the same way, who cares which company operates the bus or why?
That's not useless but it's no substitute for real time information. Seeing "Your bus is six minutes away" is reassuring in a way that "Well, the bus isn't scheduled for another minute, and maybe it's running late" is not.
In that "Oops, it's diverted" case which annoyed me, my bus was, from that point of view, genuinely getting closer, I could see it on the map. And then I realised, with growing horror, that it's on a road which won't pass me. Maybe that's a glitch? Then I saw the bus itself, in the real world, too late it's actually not coming here.
At one point Edinburgh's bus operator was part of the same legal entity as the company which provided some bus services in my city, though that is no longer true. London has it right, no tourists and almost no locals care about the bus companies. All the buses are painted the same colour, all of them work the same way, who cares which company operates the bus or why?