Had the GC survived it would probably not have extended north of Nottingham.
In the privatisation era it would presumably have sat most comfortably with Chiltern (though if it extended to Nottingham it might have needed a different name

).
The question then is what role would it serve?
In transport terms a Nottingham - Banbury service plus the Marylebone - Aylesbury service extending to Woodford would have made sense, giving Nottingham, Leicester and Rugby links to Oxford, Reading etc. (I would assume connections via XC rather than through trains)
Commercially perhaps they would "do a Birmingham" by running Nottingham - Marylebone and trying to undercut the MML operator.
Either way I fear the overheads of separate stations would put a big hole in the finances.
(But a 168 from Nottingham to Banbury would have been an interesting and useful facility.)