The track would still be state-owned, but there would be various train companies operating by regions.
My guess?
Anglia - Intercity Anglia and NSE West Anglia
LTS - NSE LTS
Chiltern - NSE Chiltern lines
Cross-Country - Intercity Cross Country
East Coast Trains - InterCity East Coast
East Midlands Trains - InterCity Midland, possibly some of Regional Railways Central Division
Great Western - InterCity Great Western, maybe NSE Thames Valley and North Downs Line services and Regional Railways South West as well
Great Northern - NSE Great Northern
Merseyrail - self-explanatory
Northern - Regional Railways North East and North West
Scotrail - Self-explanatory. Possibly sleeper services to Scotland as well.
Southeastern - Network SouthEast Kent Link and Kent Coast
Southern - Network SouthEast Sussex Coast and South London
South Western - Network SouthEast South Western Lines, Solent and Wessex and West of England
Thameslink - NSE Thameslink
TransPennine - Regional Railways Trans-Pennine
Welshrail/WalesRail - Regional Railways services in Wales
West Coast Trains - InterCity West Coast
West Midlands Trains - NSE Northampton Line, Regional Railways Centro