Cardiff bus? Although the big names are in Cardiff, Cardiff Bus has the largest number of services, and subsequently I would class as the major operator,
My list of operators shows the following cities still having major operations by council owned bus services:
Blackpool - Blackpool Transport, 175 vehicles (and 75 trams)
Cardiff - Cardiff Bus, 240 vehicles
City of Edinburgh, Lothian Buses, 690 vehicles
Halton, Halton Transport, 61 vehicles
Ipswich, Ipswich Buses, 71 vehicles
Newport, Newport Transport, 90 vehicles
Nottingham, Nottingham City Transport, 360 vehicles (17% owned by Transdev)
Reading, Reading Transport, 200 vehicles
Rossendale, Rossendale Transport, 100 vehicles
Swindon, Thamesdown Transport, 105 vehicles
Warrington, Warrington Borough Transport, 115 vehicles