Yes, an application just for London sounds specific, but I'd hope that other capital cities might be possible in future. Admittedly the data here comes from the wonderful Transport for London people and their APIs, but still... This application dates from 2017, mind you, but I'd never heard of it and I'll bet you haven't either. See below for promo text and screenshot examples.

From the Store description:

Just Buses (London) helps you find the bus stops you need to make your journey. Find the stops and services at your location, search or browse stops on a map, see when the next bus is due to arrive at selected stops and save those stops for later.

Some screenshots of this in action:

Exploring the bus stops around you, plus which routes they serve...; (right) at a bus stop, here are the routes served and the next arrival times and destinations....

Saving your 'favourite' (usual) bus stops... (right) searching for a specific stop, to find it on the map...

You can grab this in the Store here.