I have a few questions, though. Can someone on a UK passport take an internal US flight without any problems?
Yup, your passport will be used on domestic flights to check identity at security, but you won't have to go through passport controls again and they won't ask for your visa at security for domestic flights.
Also does anyone have any tips or recommendations for places to visit/durations etc., bearing in mind that I will be limited to public transport?
If you're limited to public transportation, then you'll probably want to stick to a major city. Anything outside of a major city will generally have pretty appalling public transportation (read: you need a car to drive everywhere).
You may be able to get creative with the plane tickets and routing though. I keep hearing that Norwegian Air is starting service from Gatwick to NY and Gatwick to LA for something like 250 pounds one way (on 787s to boot!). It may be advantageous to fly one of those routes or find some really cheap route combination, then try to find and purchase early another cheap domestic flight.
Something similar was what I did when I did my Europe trip a couple years ago. At that time, for whatever reason, I found it to be significantly cheaper to land in Heathrow than it was to land in Paris, Frankfurt or Berlin, so I ended up starting and ending my trip in London rather than in another city, though the locations that I would have visited themselves didn't change (just the order in which I visited the different countries changed). The cost difference between landing at Heathrow vs another major Western European airport way way more than me booking train tickets/Ryanair separately to eventually get to the same destination.
I'm not sure how easy/difficult entering Canada will be for you, holding a non-US/Canadian passport, to be honest.
Can't he just say "The Queen something something something" and the Canadians will let him in? Isn't that how the Commonwealth works?
Seattle is another city I'd love to see, but I don't think I'd rent a car simply because I haven't driven for years.
That time is just about the best time to visit Seattle, though unless you plan on doing outdoor stuff (hiking, camping, exploring geography, etc.), Seattle can be seen and done in 2 maybe 3 days max.
Also, having not driven in years? Perfect time to rent a car then. You can't be that much worse that some of the people here on the road.
On a slightly more unfortunate note, I just got word from NWU that they do not have any accommodation for me, so now I will have to rent privately or stay in a B&B for 2 months, both far more expensive and much harder to arrange. It is very disappointing, because it could mean that the whole trip is in jeopardy.
- or at least I may need to consider visiting somewhere where accommodation is alot cheaper, like India, China, or Yorkshire.
Northwestern University? I've seen it be called "Northwestern," but never NWU, and I think NWU would just confuse more Americans than "Northwestern"...
I don't have any info about rental options in and around Chicago itself, but generally, it is possible to rent for a short time for visiting scholars or short term stay people. You'll probably want to stay close to your primary campus anyways (which sounds like Evanston, but double check that you'll definitely be there most of the time as they have a secondary campus in Chicago proper). I'm not sure when Northwestern's school year is, but if you find that you'll be there during summer vacation, you may be able to sublet out someone else's lease, and that actually would probably be the cheapest option.
At worst, go on Cragislist and find a place. Though as a word of caution, I don't know anything about rental costs in Chicago or Evanston.
Anyways, if Northwestern doesn't work anymore due to cost, how about any of the other options that you talked about? Weren't you saying that UCLA (or Berkeley?) accepted you as well? Not saying other cities will necessarily be cheaper, but it's worth looking into if you can switch to another school (unless you like Northwestern so much that you'll only go there).