I'm a brit living in the US, so perhaps I can give you a mid-atlantic perspective.

Don't bother with a car in London; use the underground (aka "tube" = subway) and taxis.

UK roads are narrow and very congested by US standards, but if you are OK with driving in the evening/night, you can cover long distances quite quickly if you stick to motorways (freeways) and "dual carriageways" (divided highways). Driving 200-300 miles is not a big deal at all if you set off at ~8pm, but doing that as a "see the signts along the way" during the day is not very viable (maybe on a Sunday).

Despite what the locals will say, busses and trains are a lot of hassle if you are not used to them and don't really know where you are going. That said, for getting from London to Scotland I recommend either flying or the train - unless you have a desparate desire to experience a lot of unbelieveably inconsiderate driving.

I found that www.easycar.com had the best rates the last few times I was over, but it's worth shopping around. Be sure to get an automatic (most rentals in the UK are stick shift, but it is easy to get an auto - just don't assume...). A diesel will save you a lot on gas; they are readily available on larger vehicles. Don't be tempted to rent a small car - it will just make an already stressful experience worse; it's definitely worth paying the extra for something with a high driving position - basically a Van ("MPV", "people carrier"). Don't rent a "van" - in the UK that is a cargo truck.

I'd suggest getting from city to city by flying an LCC, and then renting locally. You will not find the LCCs on travel sites; you have to book direct. Try www.easyjet.com and www.flybe.com.

If you are looking to get the most that you can on a budget, then spend as little time in London as possible; a mid-market hotel in London can easily cost $300+. You'll find reasonable hotels outside London for less than half that. If you are looking for an American style hotel at a sane price, Holiday Inn Express is probably your best bet; OTOH Travelodge are OK, reasonably priced, and everywhere.