I thought Reliance Motor Services, Sutton-on-the-Forest, would qualify, but they only started in 1930 when they took over their York-Helmsley service from the previous operator.
Halpenny Travel, Blackrock, Co Louth, started in 1920 as "The Violet Bus Service" with a Blackrock-Dundalk service and are still going today - yes, they're based in the south of Ireland, but they enter the UK with a Dundalk-Newry service.
Northern Ireland's oldest bus operator is Yellow Line, Warrenpoint, who started-out with horse-drawn drays in 1819, before introducing a motorised Warrenpoint-Rostrevor service in 1915. Unfortunately, their business was compulsorily acquired by the Northern Ireland Road Transport Board in 1935 and - although they kept going with taxis and a funeral business - it wasn't until 1968, when there was some liberalisation of licensing, that they were able to start up again as coach operators, many years later returning to stage-carriage operation.