I took a a look at a bus route local to me on Busmiles earlier this week.

It uses OSM Carto tiles as a background map via Leaflet, but I do not think it directly uses OSM routing data or bus route relations (OSM is certainly not mentioned as a source at https://busmiles.uk/legal ).

In some cases, it draws a straight line between bus stops. In others, it does appear to follow the courses of roads in OSM, but I suspect these may be provided by a third party.