I always find it rather strange how operators outside of South East England seem to run nothing at all (or a limited special timetable) on New Years Day but yet a lot of operators in South East England treat New Years Day as a normal public holiday.
In my area most of the bus operators have always run nothing at all (or a very limited special timetable) on Christmas Day and Boxing Day but yet run a full normal Sunday timetable on New Years Day (as per other public holidays) and a normal full evening service on New Years Eve too.
Go Ahead Brighton & Hove, Go Ahead Metrobus, Southdown PSV, Big Lemon, Compass Bus, Countryliner, Sussex Bus, Coastal Coaches, Renown, etc, have always operated this way.