Hooo boy...
There are loads of places to play but there are a few pit falls that you should be warned about - at least that is what to expect in Leicester, I can't imagine it's very different in London.
Most adult teams have been together since they were 2 and are very unaccepting of newcomers. That said, if they need a player, they will let you play, and may even accept you on their team, but you will not be guaranteed playing time, even if your skills are better and you play your guts out more than any other player on the team.
Recreational soccer, is not about progressing - it is about beating the sh*t out each other on the pitch each sunday morning. It is really, really different from US rec soccer. And they will beat the sh*t out of you because you are dealing mostly with lads who probably smoke and drink way too much and don't have (or have never had) the skills to keep up. They will take you out just because they can - I recently saw an ex-Nottingham forest player's knees that have been destroyed from Sunday league, case-in-point, as well, DH's injuries playing Sunday league, neither are isolated incidences.
You will also have to face the prejudice of being from a non-football playing nation (I know, I know... football is #1 participation sport in the US, but england just doesn't see it that way.)
I really, really hope I'm wrong and that you will find a place to play, but basically, if you can avoid Sunday league and get into a different kind of team (we're just learning that there are ways around the basic thug football and there are those who appreciate the beautiful aspect of the game) then the better off you will be. It may not be in the form of formal games though, you have to think outside of the box a bit. it is available, just not in the form you expect.