I have an LNER ticket Newcastle to London on the 26th at 1130. Is it likely that some LNER services will run even if I have to wait a bit longer and/or not get a seat?
very likely - probably a normal timetable that day (or close to it).
Are there likely to be any trains between London and Birmingham late morning / lunchtime on 17th September? I know the RMT signallers are on strike that day but I am not sure whether the train drivers are.
Yes.
Only RMT on the Saturday.
And TSSA.