I didn’t book any assistance in advance as I can walk and climb steps. What I cannot do is stand for more than a minute or so without sharp pain.
OK. It's worth noting that trains are very rarely reservation-mandatory in the UK. So there is a chance (with any operator) that a seat reservation may not be upheld - rolling stock swap, computer/printer failure, severe disruption. Sadly, there are also (again, rare) occurrences where people will refuse to vacate reserved seats and train crew, unable to get the British Transport Police within a reasonable timescale, will be unable to do anything about it. There are priority seats, but these are also first come, first served. I don't know the assistance scheme well, but booking some form may be advisable to make sure you absolutely get a seat. None of this is particularly likely, but I understand you may want to be extra cautious.
92% is a great improvement! I will be traveling on a Saturday in late October. Are there any statistics about reliability (no cancellations) & on-time performance for Saturdays?
Afraid not - the stats compiled month per month. Saturdays alone would be difficult to get a meaningful data set for, as most engineering works happen at weekends. Note that there are engineering works that will affect your route on Saturday 26th October, if that's when you're travelling - trains are being diverted via Lincoln (stunning cathedral and castle if you can catch a glimpse of it). This is quite a slow route, so your journey time will already be rather extended because of it if that's your travel date (it's normally 2h). My experience of these diversions is that they can lead to extra delays, as the infrastructure isn't designed for a mainline volume of traffic.
I am also relieved to learn that I could board the next LNER to/from York if there is a cancellation. I thought LNER trains only honor LNER tickets. Does Virgin or any other rail company also accept GC tickets if a GC train is canceled?
If a GC train happens to get cancelled, what is the process of getting a replacement ticket on the next train leaving for York or London? My tickets on GC will be on my iPhone. Thanks.
Typically yes, LNER only honour LNER tickets (or inter-available tickets, of which there are many - it's a complicated business). However, in times of disruption, or cancellation of infrequent services, train operators tend to accept each others' tickets to help passengers out. Some arrangements are formal, some informal. Whenever I've seen a cancelled Grand Central or Hull Trains service at King's Cross, there is normally an announcement (and an indication on the departures board) to say that "passengers should catch the xx:xx LNER service". It's worth checking with the customer services desk if this happens, but generally the crew on the target service will be well aware of it, and you just show your original ticket.
Virgin no longer operate trains on the East Coast - from York, it's LNER to London. In the case of severe disruption resulting in the line being blocked then, all things being equal, they tend to say "tickets will be accepted via any reasonable route". So, from York, this could mean travelling via Leeds or Sheffield. Manchester if it's really bad. This is rare, and usually the result of something pretty serious.
The only exception to this if is there is such widespread disruption that another operator simply can't cope.