Honestly, this sounds like a terrible idea to me. First, as I hope you know, you definitely do not want or need a car anywhere near London. Well beyond London, yes, a car can be handy in some places (but is totally unnecessary in many other places - do your homework and decide where you expect to go before thinking about car details). If I was planning some time in an area of the UK not well served by public transit, plus some time in France also not well served by public transit, then I would simply get one car for the UK, take the train to France, and get a different car for there. I suspect that would be a lot cheaper than trying to make one car work for both, and would save a lot of hassle.
FWIW, I just got back from a trip to the UK. I did a lot of driving on this trip (drove from Salisbury to Inverness). As others have said, the first few minutes of "making the switch" are the most difficult, but I often found myself on the wrong side of the road after making a left or right turn; my wife, in the seat next to me, had the job of screaming at me "drive on the LEFT" to quickly correct me. I'm glad she was there and up to the task. As an aside, when we rented our car in Salisbury, there was another American couple who were picking up their car just ahead of us. They exited the car park and after have driven less than 100 feet, they crashed their car badly (no injuries, but both cars looked almost totaled) - this happened just a few seconds after they confidently drove off with a smile and a wave, right outside the rental agency - as we were being handed our keys, we heard the loud crash, looked, and saw the aftermath (the person about to hand me the keys to a brand new Ford rolled her eyes and gave me a hard look - I said "don't worry, I won't do that!"). Fortunately, I ended up driving almost 1000 miles, and never had an accident (although my tires did bump the curb at high speed too often - see below).
A couple more notes: on the motorways, the driving "on the wrong side" is easy enough. The roundabouts do take a little getting used to (some are easier than others). But I found the narrow-ness of the roads surprising and very challenging (most of the roads, not just out in the hinterlands). I also found judging the distance between the far side of the car and the edge of the road extremely challenging - I occasionally "bounced" off the curb, while driving along at high speed (because I couldn't easily judge how far the left side of the car was from the edge of the pavement, and the oncoming traffic seemed awfully close). It was very disconcerting. Then there were the single-lane roads with occasional "passing places"...those I actually found kind of fun.
Oh, and about those speed cameras: the warning signs about speed cameras are everywhere - I dont think we drove more than a few minutes anywhere in England where there wasn't a sign about speed cameras - it was incredible (not so many of them in Scotland). I don't know how many actual cameras there were (we saw a few) but the signs seemed to be pretty much on every mile of every road. No speeding tickets that I've seen (yet - it's just been a week or so...we'll see; I did try hard to obey all speed limits, but none of the cars around me seemed to be).