When I visited England with a friend in 1976, my experience was similar to Powers106 and Encinitas.
I’ll add that on two occasions, when passing on a two-lane road, it took me one or two seconds longer than usual to realize I had to move back into the left lane! No near-misses or anything serious, fortunately; just a little excitement. 
Funkynige posted a link to the so-called ‘Magic Roundabout’ in an English town called Swindon. I’ve actually driven around this and it’s not as horrific as it looks.
In any case, don’t worry about it. Swindon is the only place that has one of these monstrosities and visitors to these shores are highly unlikely to have any reason to go anywhere near Swindon, let alone actually drive around it. It is a fetid dump of a town with no redeeming qualities at all, no culture, no industry, no life, and entirely populated by brain-dead deadbeats with the mental life of a ceiling tile and all the creative zest for life of a window ledge.
I’ve driven in the UK a bit. I want to know what popular UK car has the 1-2 action closest to the driver. Because I’ve never seen nor heard of that.
Oh, and those “Look Left” and “Look Right” signs for the “dumbass Americans stepping into traffic looking the wrong way” - they have them in Spain and France too, albeit in the native language, and I’ve even seen them in Poland. And they all drive on the same side as those “dumbass Americans”. Hmm…
BTW - the Magic Roundabout is the Plough Roundabout in Hemel Hempstead, thank you very much. Six, not five like Swindon, interlinked roundabouts - and I mastered it very well - ask Fierra. 
Five roundabouts - what wankers. Real women do six!
The advice I got was “Put your passenger in the ditch.” It seemed to work pretty well. You learn to plan your way into and through the next intersection, and to be wary of parking lots, single-lane driving and other non-standard situations. Most people catch on rather quickly.
As a testament to this, consider that few countries (none that I know of) put restrictions on drivers who come from countries where the driving is on the other side of the road. I’d say that’s evidence that most drivers are able to handle the switch.
I’ll support the view that learning to cross the street is harder - you should make yourself look both ways twice. And the hardest thing of all is learning how to climb into the correct seat for driving. Long after you’re confidently handling roundabouts you’ll find youself sitting in the passenger’s seat, cursing the lowlife that swiped the steering wheel.
Once you are well adapted, take care for the first few days after you return home.
A trivia question: Where in U.S. territory is it the rule to drive on the left? (Note that this area was never under British control.)
I drove from London down to Salisbury and then up to Stonehenge and back to London without too much trouble.
One thing I did discover and it’s worth watching out for. In a tight spot your reactions are backwards. For example, to avoid an accident in the US the tendency is to pull to the right. That’s exactly wrong in left-side drive countries.
I discovered this when we were driving along nicely on the way. The two women in the car suddenly said, excitedly, “Oh look, thatched roof cottages!!” The scene was on the left side of the road. Instead of simply pulling off to the left onto the shoulder, I went up to the next intersection, turned around when it was clear and came back on the wrong side of the road!
The other thing that took a little getting used to was passing. I forgot that most of my car was on my left and I tended to cut in too soon. But the other drivers took care of me.
In towns following other traffic helped. On the highway I had to guard against complacency.
Just a wild guess: The Virgin Islands 
I realize that a possible response here is “Various streets and sidewalks in Boston.” But that area was once under British control.
kniz takes the kewpie doll. In the USVI they drive on the left (generally in LHD cars). I was told that this started when the islands were under Danish control; Denmark later switched.
The fun thing about roundabouts such as those in Swindon and Hemel Hempstead is that you can legally navigate the large roundabout either clockwise or anticlockwise. You go clockwise around the outer perimeter or anticlockwise around the hub. Simple really!
On the subject of looking the wrong way when crossing the road:- This once led the the arrest of a British agent who had been landed in occupied France during WW2. One day he was waiting to cross the road and looked the "wrong " way when standing on the curb. His action was spotted by an alert German officer and he was taken into custody.
I made the switch from US to UK driving in two steps. When I moved here, I brought my American car with me, so I just had to get used to driving on the wrong side of the road, which wasn’t too bad. I then bought a right hand drive car and I found getting used to being on the wrong side of the car was harder than being on the wrong side of the road. I kept driving too close to the left edge of the road and clipped a couple kerbs.
I’ve now driven all possible combinations of left/right driving and steering (American cars in US and UK, British cars in UK and France), and after the first couple minutes it becomes natural.
I humbly retract my earlier statement and blame in on a faulty memory :(, which was probably out to lunch at the time, driving an automatic car and trying to stay on the right side of the road (or the left side, depending on where it was having lunch that day).
Sorry, all, for any confusion!