What about the, er, LNW-GW Joint between Chester and Warrington with GWR running powers onwards via Earlestown to Manchester Liverpool Road?
Yes, you're right of course, but there was no significant GWR property in the north west (except goods depots).
The line was built as the Birkenhead, Lancashire and Cheshire Junction, and became the Birkenhead Joint.
The GWR certainly worked over it to Birkenhead, Manchester and Liverpool.
And also into Crewe from Wellington via Market Drayton.
The Great Northern worked into Manchester Central of course via the MS&L/CLC lines.
It's odd that "Great Northern" is a very prominent name in Manchester today, based on its long-time display on the GNR goods warehouse at Deansgate.
Albert Dock in Liverpool has a prominent GWR sign on the old goods depot.
The Midland still has its hotel in Manchester.
The LNWR is sadly relegated to a sign on their goods warehouse at Heaton Norris.
All the other references seem to have vanished, although Network Rail keeps the flag flying with its LNW Route appellation.
And not to forget Wigan North Western.