The U.K. in 22 Days

Matt, my wife and I are doing nearly the same in Mid-September. We're not going to Scotland but we are going to most of the places you mentioned. We plan of going to Bath directly from Heathrow and beginning there. We'll day trip to Stonehenge, Oxford and Westminster from there. We'll also be there for the first weekend of the Jane Austen Festival. Then, it's two days in Chipping Campden (Cotswold) with a day-trip to Stratford-On-Avon. Then, it's on the Coventry for a short visit and then onto the South Lakes and then two days in Keswick. Then, two days in York. One night in Cambridge (We like Universities) and one night in Cantebury. Finally, 6 nights in London. We're renting a car in Bath for 12 days and turning it in when we get to London. With full CDW, we're looking at 42Euro/day.

If you PM me with your email ID, I'll be glad to share our Excel planning spreadsheet that shows our 20 days and nights. (We'll share with anybody who wants the spreadsheet).

between cities prebooked train tickets would be far cheaper than a car.
http://www.nationalrail.co.uk/

I've just found this site also,
http://railsaver.co.uk/index.html

Megatrain/bus do rock bottom fares(limited)
Vaguely from the south to the north...
Chilton rail do a good deal to Stradford upn avon.
Mega train can be £5.50 Birmingham-Edinburgh
From Manchester theres "sail and rail" ticket , if you want to hit Dublin for about £30.

Rail rover area tickets
http://www.guardian.co.uk/travel/2009/feb/03/rail-travel-budget-england-uk
The Bay-to-bay walk , at Morecombe might be an option.
Theres a scotrail train pass too, the coast of the north west is amazing.
If you are outdoorsy, there quite a few bothies in the highlands.search "M.B.A"
Lots of independent hostels here too, and "wigwam" campsites.

Have fun planning, but don't spend all your on the road

Consider this: London- 5-7 days, day trips to any of the following Canterbury and Dover, Oxford, Bath, Stratford-upon-Avon/Cotswolds, possand Windsor, possibly to Cardiff Wales. Train to York- 1 or 2 nights. Train to Edinburgh- 2 or 3 nights. Train to Aviemore- 1 or 2 nights. Train to Inverness- 5-7 nights. Day trips to Orkney, Isle of Sky, Great Glen or Glen Coe, Culloden battlefield and Clava Cairns, Whisky Distillery tour (here or Aviemore). Train to Sterling 1 night. Train to London- 1 night. We've done similar trips- though not quite as long. You'll love it! Feel free to Private Message me- we've been to the UK more than 40 times.

Matt, I'm going to second Toni's advice on adding a bit more of Scotland if you can. I think you'll both like it. If you find that you can't get all the way to Inverness you might want to think about taking the train to Birnam (yes that Birnam) which will give you some idea of the highlands. You can see Dunkeld and hear some Scottish traditional music at the Music Bar. Stirling and Linlithgow are easy train trips from Edinburgh. Both are great easy to reach destinations. Now, you might want to think about Glasgow. Both Edinburgh and Glasgow have large student populations. Glasgow has a lively music scene if you are interested.

Pam

Also check national express for coaches.They are by far the biggest network, and have limited "fun-fare" tickets if prebooked. Just found £1.50 fare London-Edinburgh

Couple of amazing scottish hostels, loch Ossian -out in the wilds.
And, http://www.carbisdale.org/thecastle.htm.
Don't know the answer, but a student railcard would get you 1/3 off all rail fares for a sign up of £24.I'd assume with some US student ID card you'd qualify.
The railway journey up to Fort William is wow, it passes beneath Ben Dorain , and drops you at Corrour Halt.
If you make Aviemore, the little shop by the forest holidays campsite , has a sauna for a few £.Nice after a walk to the Blue Lochan.

Look at Hostelworld.com for places to stay and prices. You might want to book your first couple of days and last couple of days in advance. Easyhotel.com is a cheap hotel in London. NO frills, everything costs extra but if you can do hostels then this should work.

Have you thought of open-jaw tickets? Would save back tracking. Where you are flying home from is where you should visit last. If that be London, then do London last, not first. That way, you are where you are leaving from early and not trying to get there.

The arriving day is screwed up anyway. Jet lagged and tired. Have a train ticket or bus ticket or cheap airline ticket and go north to Scotland first. You can fly to Edinburgh or Glasgow or Aberdeen. Bus or train will take you most anywhere. Then work your way back, assuming you are flying home from London.

Car rental, if you can drive a stick, is fairly cheap. Gas, however is probably over 8.00 per gallon by now or even more. In Scotland, it might be worth renting for a couple of days to go and see the highlands. Bus and trains are sparse there.

I remember that when we were in Edinburgh, it was cheaper for the two of us to take the bus to Glasgow than it was to pay the gas for the car. Look at your itinerary and see where a car might be used. If it looks like it can, then rent one, if not, then bus or train.

I think you would like Inverness also. If you go, look at Johnny Fox pub for average food but great bands at night. A good place to stay would be
Ardgarry B&B -http://www.ardgarryguesthouse.co.uk/accommodation_tariffs.html
would be around 44-50 pounds for 2- per night, with great breakfast.

There are decent Hostels there with rates from 20L per person. Be sure to check the rating of hostels before you book. Would also book private if possible at hostels. My son did not and left a hostel with 5 days left on the bill and found another place to stay. It had a bad rating, but he did not do his research before he went.