You can change the name in your passport via the US Embassy in London. It might take a week or two, and if your passport is more than a year old, it will cost you. You don't have to change the name, though, if you don't want to. You can carry a copy of your marriage certificate with your passport, and that will be sufficient to show your name change in the UK. However, if you do choose to change the name, definitely do it before you apply for FLR(M) because it will cost you £160 to move your visa vignette from one passport to another (although now that I think about it, they're giving ID cards for FLR(M) holders now, so that might not be an issue). It's your choice, really, there are pros and cons to both options, it depends on how important changing the passport name is to you.