I moved my horse from the US to the UK and it's expensive and a hassle, but not impossible. I don't regret it at all. The flight cost $4000-ish but it definitely adds up when you have to move the horse from somewhere like Arizona to the East Coast before it even gets on the plane. Only a few airports handle livestock -- Miami, LAX, and JFK. That's it as far as I know. So the horse will have to travel overland to one of those airports. I think Stansted can take horses but in our case, it was cheaper to fly to Schiphol (Amsterdam) and then travel overland to the North of England.
The horse had about ten days worth of travel. It was three or four days to New York, then she had a layover for a few days at a barn Upstate. Then to JFK and on the plane. I met her at JFK, got to hang out with her for a couple hours in the stable at the airport, and then we flew in separate planes to Amsterdam. As far as I know, she had a lovely flight. Mine sucked. I got stuck in Frankfurt because I missed a connecting flight, my bags got lost, they nearly didn't get me on the connecting flight. But after all that drama, I made it Amsterdam and we were reunited and then we stayed overnight at a barn near the airport (try telling the lost baggage people in Schiphol airport that your overnight accommodation in Holland is the "Horse Hotel." No, really, that's what it's called. They look at you as if you have totally lost your mind). Next day, we were on the lorry (truck in American) to England. Took the ferry from Calais and spent about three days at a barn in Kent waiting for our lift north. I was at least able to ride her a bit in this field near the barn but otherwise we just sat around looking at each other, wondering if this really had been a good idea. Anyway, our lorry arrived and we went as far as Newmarket, which is kind of in the middle of the country. The horse had four-star accommodation near the racetrack. I slept on the lorry. We were off at 6am the next day on a rambling meander through England, picking up horses destined for Scotland. That must have been about eight hours of driving. We got to Durham in the end. Other than my flight debacle in Frankfurt and traveling across Europe with the clothes I was wearing, a horse, and nothing else, as my luggage had to travel to Durham separately, it all went very smoothly. It was stressful on the horse but not unduly so. I was probably more traumatized. :) By the end, she was probably thinking, "Oh, God, not another bloody trailer" while I was thinking, "What have I DONE? I've just moved myself and my horse to this crazy foreign country!"
I dunno...maybe that gives you a sense of what it's like. You don't *have* to travel with the horse. Most people don't. They just meet the horse at the other end. I just thought it would be easier on her if I did.
The only other thing.... you say "London area." Does that mean IN London or somewhere near the M25, like Essex? If you're in London, you'll have a hard time keeping a horse unless you can spend a lot of money doing it and are willing to do a lot of commuting. But I think you'd be fine somewhere more rural outside of London. I'm only speculating based on what I know of London (I've visited it a few times) and know of keeping the horse on Staten Island, briefly, when I had a summer job in New York City. Different sort of cities of course, but similar populations and similarly epic urban sprawl. But I've never lived in or near London, only the northeast of England and Scotland, so maybe someone on here who does can better advise as to what that's like.