2 week itinerary

Yes, it is physically possible, but you would not have time to spend to really visit very much. I commend you for spending 5 days in London. After you visit York and Edinburgh; the trip via the Lake District and the Cotswolds would be more like a preview for a later trip. We did a similar trip in 21 days. If you have a Scottish ancestry, you might want to spend more time in Scotland because there really is a lot to do and see. See the Visit Scotland and Historic Scotland online sites. If you want to see Ye Olde England spend more time I the South and the Midlands. See the National Trust UK oneline site for itinerary ideas and info on Overseas Visitor Pass.

Fourteen nights is tight. I would do something like this ...
London - 4 nights (sorry had to sneak a night off here)
Train to Oxford. Pick up hire car for Cotswolds - 2 nights
Drive to Peak District - 1 night
Drive to Lake District - 2 nights
Drive to York - 2 nights (return car)
Train to Edinburgh - 3 nights
Fly out of Edinburgh

Two weeks is sufficient time (just) to see London 5 nights, York 2 nights, Edinburgh 4 nights and back to London for your final night, assuming you are not flying open jaw. Travel by train.

You need another week IMO to cover the Lakes and the Cotswolds. Don't forget that there is a Bank Holiday here the last weekend in May, when places get busier and the traffic even more hideous than usual!

You had some good suggestions from earlier posters, but frankly, with two weeks and three great cities to visit, I would skip the car rental for this trip. Plan on doing that on another trip.

Driving from the Cotswolds to the Lake District will use up most of one day of your valuable 14 days.

We did two weeks with four nights in London, plus two each in Bath, Keswick, Hadrian's Wall (Once Brewed) and a day trip to York. It was a whirlwind tour and it was all by train. And we're two adults in good walking condition with no kidlets in tow. Your mileage may vary.

I did something like this quite a while ago (17 yrs!) while backpacking. We spent 4 nights in London, then rented a car and hit Stonehenge on the way to Stowe on the Wold(2 nts). Arriving in the evening, we spent the next day exploring the village and environs by taking nice walks. The next morning we drove straight through to Glencoe and spent two nights there. We walked the loch and climbed the pap. We drank whiskey with the town historian, Arthur, a Rick Steves legend, now passed :( and then took a day trip to Fort William, up Loch Ness and into Inverness. We stopped at Culloden and at Clava Cairns, which were not well known at that time. We continued our drive that day to Edinburgh for two more nights. It was a long day of driving, but it worked. I remember visiting the castle as well as Stirling Castle, and the Royal Mile. We then drove down to York, spent one day there, stayed overnight and then drove back down to London.

11 nights total. It was too quick, but we really filled every moment and got a taste of a lot of different things! So, is it do-able? Sure. Is it better to have more time? Absolutely. But it can be done.