Help some English Guys RV Across America (please)!
Oh Nick, where to start...
Is $5000 all that you can put up to pay for one, or what you are willing to loose between buying it and then reselling it on the other side of the country? You'll have a tough time finding a car good enough to cross the country in, let alone an RV if that is all you are willing to pay to purchase it. As far as $5K being the loss between buying it and dropping it off, perhaps it might work, depending on how you sell it.
As with any used vehicle purchase, the older the vehicle you get, the more likely you are to have mechanical problems. With a motor home, you have two sets of potential problems--vehicle problems and then "house" problems. How mechanically-inclined are you, and how tight is your schedule?
Also, exactly what do you mean by "RV"? We have campers ranging from units which fit in the back of pickup trucks and small tent-trailers, to converted tour buses. Obviously, the larger and more extravagent you go, the less realistic your budget is.
Then you have to register it and get license plates. Without a local address, that might be a challenge. You could form a US corporation and register the vehicle in the corporation's name, but the legal fees will eat up part of your budget, not to mention the time delays. Insurance will be a challenge, as the US has their own system of sharing information between states, but not internationally.
Since vehicle registration is a function of the states, there are 50 different sets of regulations. I'm sure with the post 9/11 (precautions/xenophobia), it hasn't gotten any easier. This might be a good question to ask the British embassy in Washington, or the US embassy in London. If you are serious about pursuing this, you should also look into buying/registering/insuring it in Canada. Their laws might be be more agreable to your situation, with the Commonwealth and all.
Are you sure you can rent one for $5000 for a cross-country trip? By the time you figure in the daily charge, how many miles they include, and even if they will let you drop it off across the country (be sure to ask about drop-off fees), I'm not sure (although, I'll admit, I don't really know...) if that is doable. Lay out a planned route, and calculate the mileage (
www.mapquest.com is a good way to do this). Most people who rent their RV, drive a loop, and drop it off back a week or two later where they picked it up. Check out the national RV rental agencies, like
www.cruiseamerica.com about cross-country rentals.
One twist you might be able to use for your plan is picking the RV up at the factory, and then driving it to its new home at the rental agency. I know that some agencies up here in Alaska have special plans like that to get their new units delivered at the beginning of the season. By the way, most RVs are made in and around Elkart, Indiana (about 100 miles east of Chicago). Check with RV rental agencies at and around your planned end point and ask if they do this.
...Then there's your time frame. Your camping options will be driven by regional seasonality--you'll find that most campgrounds in the northeast and north-central part of the country will be closed, due to snow and freezing temperatures. In MOST RVs, you need to drain the water systems if the temperature drops below freezing, so you'll lose the use of your RV plumbing systems (there are all-season RV models, but these are not typically at the low-end of the price scale). So, unless you intend to cross the country way down south, you'll be stuck spending the night in parking lots or roadside rests, wishing you were in a hotel! Many of the national parks I presume you'll want to visit are closed in the winter (Yellowstone, Glacier, etc.) due to snow covering the roads--so inquire as you plan your route.
Instead of camping, you might want to instead consider renting a car and driving cross-country but staying in hotels. There are plenty of descent lower-priced hotel chains (like Super 8, etc.), which are good if you are on a tight budget. Or buy a tent and car camp out of your rental car (again, down south).
Wish I could be more supportive...