Most places seem to have no service on New Year's Day. A few years back trentbarton ran a limited service on New Years Day but only on the main routes (indigo, rainbow one etc) but the boxing day service was also limited to main routes on special timetables rather than the Sunday service like last year.
Oddly, many places in the South which don't have the best bus services in the country do have a Sunday or similar service on New Year's Day.