There is a significant proven demand for local transport in cities on 26th December, for shopping and sporting events mostly.
A limited local bus service is usually offered to support this in most places, and light rail services and London Underground usually run.
A significant number of local trains do run on Merseyrail, to and from Glasgow and London Victoria, though some of these are bus replacement today. Chiltern Railways trialled something similar last year on the basis of serving shopping in Central London and at Bicester Village. It is being repeated this year apparently with financial support from the shopping centre owners.
Basically, local public transport services which are demand-responsive do offer a service on Boxing Day except national rail and the above exceptions.
The railway is however choosing to offer a comprehensive local train service in England (except the North East) on 1st January despite there being fewer potential customers for it than on boxing day. New Year's Day is not nearly as established as a big shopping or major sporting day that I am aware. In Greater Manchester there are many commercial bus services running on 26th December and no trains whatsoever. There are almost no buses running on 1st January but a full weekday timetable of local trains is running. This gives you some idea of how out of it the railway industry is. As far as English local trains are concerned, it would be an improvement to flip the situation, so there is a very limited service on New Year's Day but a good service on Boxing Day. It's not like early morning service, high frequencies or peak extras would be required on either day.