I'm now in the slightly embarrassing situation of having to admit that I've never been to Scotland! I know, I know, lived and traveled around the U.S., been to France, Spain, and so on, but never set foot in England's next-door neighbor to the north.
Anyway, as you've said, you're not going to find anywhere as deserted and unpopulated as much of the U.S., certainly not once you make it down as far as the Midlands and South of England. Out here in East Anglia we have a lower than average population density, but places will still seem pretty close together coming from America. Once you start getting nearer to London, expect congestion. Every time I go back down there it seems to be worse than the time before. To be honest, I find it a rat-race now.
Anyway, I'll second the suggestions about renting a car. The rail network will take you around to some places, but if you really want to get off the beaten track and explore the countryside, a car is just about essential. Britain has fewer rail branch lines now than in the past, thanks to the (in)famous Dr. Beeching and the government closing and tearing up many of them during the 1960s/70s.
I don't think you should find driving on the "wrong" side that much of a problem after a few miles to get used to it. I guess some people find adapting easier than others, but having driven all four combinations of LHD and RHD cars on both sides of the road, I've never had any problem switching between them. Negotiating roundabouts (traffic circles) -- of which there are many in all shapes and sizes -- will probably be the point where you need the most concentration, watching for approaching traffic and remembering to going clockwise into the circle.
Other important driving rules while we're on the subject:
* At roundabouts you must yield to approaching traffic from the right.
* Speed limits are 30 MPH in town, 60 on the open highway, and 70 on divided highways ("dual carriageways"), unless otherwise posted. Once you reach any main highway these days, speed cameras are everywhere.
* Turns on red are
not permitted anywhere (unless a green arrow is displayed simultaneously).
* "No passing" lines follow the same solid/dashed principle as the U.S. but are
white, not yellow.
* Yellow lines at the edge of the road indicate parking restrictions. London and some of the other large urban area also have red lines in some places now. Don't even
think about parking there.
Maybe I should put an American Flag on the front bumper so they will know I might turn the wrong way.
I have one on the back of my little Bronco II: