I absolutely love London.
I've thought about making the jump across the pond again, but I'll tell ya that London isn't the most expensive city in the world for nothing. My colleague's sister is actually moving out of London to sunny old Manchester because of this.
I've always thought of London as an island; it's an oval-shaped city, surrounded by a freeway (motorway in Brit-speak) called the Orbital. Outside of the Orbital are the counties of Sussex to the south, Kent to the southeast, Essex to the northeast, and I forget the rest of my London geography. In between London are boroughs or districts like any big metropolis you'll find in the world, each with a different vibe; some are more ethnic communities. All the tourist attractions (i.e. Parliament, Tower of London) make up what's called Central London. The West End of London tends to be more cosmopolitan and affluent (Chelsea and Kensington for example), kind of like Broadway in New York you could say, whereas the East End tends to be more working-class (West Ham, for instance), but the same can be said for parts of North and South London.
I don't know much about apartment searching in London, but I reckon you'll find a vacant flat wherever you go. You could consider living outside of London and driving to the subway depot on the fringes of London and take advantage of the park and ride there (it's called a car park not a parking lot mind you). That could be a cheaper way of settling in. My English uncle lives in Essex and this is how he commutes.
No matter what side of London you live in, the city feels a lot shorter when you take the Tube (the London subway system). It's easy to hop on and off in between lines as all the route maps are color-coded. The waits are very minimal (the longest I recall waiting was a minute and a half). Very efficient system. The buses are quite good too. You can find a subway depot on about every other corner. I think it's quite difficult to be lost in London or even any other European city for that matter given the availability of mass transit.
If you must drive, remember that your vehicle is right-hand drive and you drive on the left. You will run into an annoying circle called a roundabout, which basically is England's WTF version of an intersection. Make it a habit to yield if a car wants to cross your path in order to get out of the roundabout and onto another road. It's confusing and so, you'll have to learn on the fly while you're there. You better be good at driving stickshift, BTW as most vehicles there have manual trannies. The penalties for traffic violations are more severe in London also.
As for transport to the EU, you can ferry your car over to say, France and Holland, or take the train. The Eurostar high-speed train will take you to the continent, but you'll have to book months in advance, especially if you intend to go to the continent during the height of summer.
Soccer is religion in England, but I think London is quite a laidback place for the sport (namely Arsenal and Chelsea), unless you ventured into the parts of the city that are hubs for West Ham and Millwall. There's a saying that the farther north you go in England, the rowdier the soccer fans. I advise you don't call it soccer: it's football or footie (just to sound cute I guess). I also advise that you don't wear a club shirt in a neighborhood where that club isn't well received. For example, wearing an Arsenal jersey in a pub in Tottenham is NOT a good way to socialize, as Arsenal and Tottenham are heated North London rivals. By the way, I guarantee you if you stick around England for quite a long time, that you will find a club to root for...and often times, it's the club that chooses which team you patronize. As for me, I love soccer, as I grew up watching it, but as a Manchester United fan, I have to fly under the radar a bit as far as where my allegiances lie.
BTW, you better get used to cricket. Yuck. You think soccer is boring? You haven't seen cricket. I remember India was walloping England by 300 runs over the course of 4 straight days and yet, there are folks who will watch some lopsided rubbish. It's just a game I'll never understand.
What to bring across the pond? I suggest that you buy your electronics in London, unless you have a conversion device that converts American voltage to European and vice versa. The outlets are distinct from ours here. I guess you should bring your clothing with you, unless you want to pay a bundle at the boutiques in London. They don't charge you sales tax in England, but I think it's tacked on in the markup anyway.
Well, I hope this helps. It's a huge adjustment and England isn't great for claustrophobics because the homes tend to be smaller and are bunched up together (terrace houses) with little or no front yard/backyard. Food can be quite expensive (of course), even from the American franchises and the portions are smaller. Get used to talking another kind of English too because American English is quite different from English English that you could swear they're different languages. But hang in there. It had taken me a week or two to get used to life in Europe, but could be longer for you if you've never been used to it at some level.
BTW, the girls in Essex have a reputation for being easy.