Avanti share their route entirely with other operators.
Euston - Birmingham/Trent Valley - Liverpool with LNWR
Coventry - Birmingham - Crewe with XC
Crewe /Stoke - Manchester with XC & Northern
Crewe - Chester & beyond with TfW
As stated above Wigan / Preston - Glasgow / Edinburgh with TPE.
Wigan - /Blackpool Oxenholme with Northern (albeit different Northern services).
Euston - Glasgow / Edinburgh with Caledonian Sleeper
I'm sure there are more with Avanti alone.
LNER share their route with:
GC from King's Cross to Northallerton
HT from King's Cross to Doncaster
Great Northern from King's Cross to Peterborough
EMR from Peterborough to Grantham, Doncaster to Leeds, and Doncaster to York
Northern from Doncaster to Leeds / Skipton / Harrogate
XC from Leeds/ Doncaster to Aberdeen
TPE from Leeds to Edinburgh
Northern from Darlington - Alnmouth
Scotrail from Dunbar - Aberdeen/Inverness
Caledonian Sleeper from Edinburgh - Aberdeen/Inverness
TPE only are the sole operator on the line between York and Seamer (for now).
Everywhere else is shared with at least one of Northern, LNER or Avanti.
Hull Trains shares its route entirely with LNER
GC shares its route with LNER, TPE and Northern on the Sunderland leg, and LNER and partially northern on the Bradford leg.
Merseyrail, as far as I am aware do not share their route with anyone. I'm not sure if any other TOC is in this position?
XC shares its route in far too many places to count. It shares its route with Scotrail, Avanti, Caledonian Sleeper, LNER, TPE, GC, HT, Northern, EMR, WMT/LNER, GWR, SWT, AGA, TSGN, TfW. That's 15 ToCs and I presume that is the maximum.