Cotswolds and Cumbria.

Bath is south of the Cotswolds proper though not so far as to make a trip down problematic. However, to save time you might consider having a day there first, driving north to your Cotswolds base, having 2-3 days exploring, then head north. The drive up is long and not that inspiring for the most part as it's all motorway (M5 then M6) although once past Preston it's not too hideous. You could theoretically do the trip in about 4 hours if the traffic was very light going through the west midlands and past Warrington, but you'd be pushing on hard to do it; allow 5-6 hours so you can make coffee & lunch stops. The train from Bristol takes at least as long once you factor in time to drop off a hire car in Bristol and pick up another in Keswick. From there the M74 is an easy drive up to Glasgow and on into the Highlands, or you could wind your way up the A7 to Edinburgh. Have you looked at going straight up through Glasgow, then at the end of your Scotland leg head back south through the Cairngorms & Perth down to Edinburgh? Saves a lot of zig-zagging around that way, especially if you use Edinburgh airport as your departure point...

Brendon: The Cotswolds is very charming with many historic homes and antique stores. The countryside is very very green but not very dramatic...think Thomas Kincaid. Bath is a perfect example of Georgian Architecture. Stratford the town isn't much but you can attend a Shakespeare Play. The Lake District caters to the daytrippers. Cumbria is more starkly beautiful; maybe more what you are looking for. I would recommend spending more time in Southwest Scotland and the Borders before racing up to Edinburgh. Glasgow has some wonderful art and history museums. Edinburgh has the Castle with a huge military collection. If you travel linearly you can see a lot on the way. Carry a guide that lists by area so you will not miss something that would be exactly what you were looking for. Bon Voyage

How many days do for have total for England? Just three before you go to Scotland?

We drove from east London to the Lake District and it took 4.5 hours including stopping for a quick breakfast. It should be about the same from the Cotswolds.
I've spent a great deal of time in the Lake District and can suggest the following things to see...in no particular order!
Buttermere - one of the loveliest stretches of water I've ever seen. You can walk around the lake in about two hours and then have tea in the town of Buttermere. The town is a little busy in summer but well worth seeing. Drive from Keswick up over Honister Pass...stunning.
Aira Force - beautiful waterfall
Castlerigg Stone Circle - above Keswick. Sweeping views of the surrounding fells and farms and one of the oldest stone circles in England.
The park at the end of the lake in Keswick - hike up the grassy hill overlooking the lake and stay for sunset.
Langdale and the Langdale Pikes - beautiful Lakeland village and surrounding hills.
Walk around Rydal Water
I could go on and on but these are a few suggestions to get you started!

No planes fly from the USA to Bristol apart from some taking Brits to Florida. So, it looks like you will land at Heathrow.

From a Cotswold base, it is easy to also drive to Stratford-upon-Avon and SE to Blenheim Palace. If wishing to go by train to Scotland, then Cheltenham would be a good place to depart the Cotswold area.

There are Flights from SEA to Bristol with stop in Amsterdam. A friend of mine who is English flies that way to go home. It is so much easily for check-in and check-out.