Seeing the rest of England in between London weekends

What you've sketched out does not meet my definition of a relaxed pace. Four destinations in five days, north, south, east, west - - one might say it's all over the map. Another consideration is weather. Rain is rather likely in April, and I don't think there's much to do in the Lake District in that case. York, on the other hand, has plenty of indoor attractions. I suggest you not make any non-refundable reservations. See what the weather prediction is like a day or two ahead, then make your decisions

Head straight to York and pick up a rental there ...

Sunday (am) - Train to York (1nt)
Monday - Pick up hire car and drive to Keswick (2nts)
Tuesday - Exploring the Lake District
Wednesday - Drive to the Peak District (1nt)
Thursday - Drive to Cotswolds (1nt)
Friday - Drive to Taplow

Most of your time is in the north so I'd concentrate on the Yorkshire Dales and the Lakes and skip the Cotswolds.

Which side of London will you be on?

There are frequent trains from Kings Cross to Cambridge.

I agree that it all seems a bit rushed but maybe you like it that way.

I hope that Cumbria gets back on its feet before your trip. You may be aware that last week there were horrific floods from Storm Desmond and even today there is new flooding.

I'd check before driving there.

Unless you have to drive through the centre of London - even on a Sunday for a new driver - I'd consider driving.

Thanks all for your advice! Agreed, it was a little too rushed, even for us. We're going to drop the Cotswolds and due to some flight schedule changes we're now going to have an extra day in London before we leave for the country side. Thus, we're now thinking:

Day trip to Cambridge during central London stay.

Sunday- pick up rental car at Heathrow. Drive the 4 hours to York. See Castle Howard before 5. Drive to Yorkshire Dales for evening. (Thanks @ramblin' on-they look great)

Monday- Yorkshire Dales National Park. Use Grassington as a home base? Nights in Grassington or other home base. Suggestions?
Wednesday afternoon - head to Lake District, Keswick as home base. Since we have a car, is there a better base than Keswick?
Friday - head to Taplow for family shindig

Does this sound better? Aside from the unavoidables, like rain or town still recovering.

Thanks again for the input.

You asked if Keswick is a good base with a car and it is. Parking is far easier there than most of the other towns in the area and it's easy to get everywhere you want to be. If you combine York and the surrounding area with the Lake District you will have time enough for an overview of both. Adding in the Cotswolds would be far too much!
Cumbria will be back on it's feet by the time you get there. The British army is currently fixing the A591 as well as several other roads in the area that were taken out by the incredible flooding. The two main grocery stores in Keswick are currently closed but will be opened again in several weeks. In the meantime they are bringing in pop-up stores to help keep the town going and the local shops are also adding stock to insure that everyone has what they need. All of the towns in the area are in clean up mode. Most businesses are back up and running. They really need the tourist dollars right now...apparently the place is deserted of visitors even though most things are back to normal. I wish I could go right now myself! I'd love to be in the area when there are no other tourists around!
If you're on Facebook head over the the I Love the Lake District page for up to the minute info and postings.