York is absolutely wonderful - no argument. The east coast train line is fine - sure. However the OP wants to hit Oban, the Lakes, Knutsford, and Broadway which are ALL on the west side. Fitting in York would mean eliminating one or more of those. They seem to know where they want to go . . .

You are great! So helpful that you instill confidence that this is doable. We well know that it is impossible to cover a fraction of what is amazing about this beautiful country. Your info just gets us more hyped up and excited about going. Every time we've traveled abroad, we've regretted not having more time to visit wonderful places. We live in Northern California and run into tourists that express the same desire to spend more time in Yosemite, the Redwoods, etc. Well, we can always come back. Thank you all for taking the time to help us enjoy England and Scotland.