The plan is perfectly doable without hiring a car.
London to York is an easy 2 hour train ride, regular service from King's Cross station. Buying the tickets now will get the best prices, especially as you can fix your train times now, which allows you to buy Advance tickets. Tickets are available now for the evening of Monday 13/11 for a little over £50pp.
There are plenty of options to explore for you whilst your husband works. You need a day at least to see York itself. then you may be able to find a tour that will let you see either the Yorkshire Dales NP, the North York Moors NP and the Yorkshire Coast, but at that time of year they may not be running. have a look at Mountain Goat tours.
Durham is an easy day trip by train. Edinburgh also doable but a much longer day. Manchester is a less well known destination for overseas tourists but has lots to engage you and is an easy day trip.
The Lake District is again doable for your time after York, although it's a bit more awkward to get to by train from York and November is not the ideal time to see the English countryside with possible poor weather and definite short daylight hours. Manchester and Liverpool would both be good city break locations, especially if you can fly home from Manchester Airport, saving the return to London.