Depends on the age at which you move to UK. If before puberty, you are very likely to pick up the accent of the people around you. After puberty, your accent gets more and more fixed, until adulthood, when you will always retain some of your original accent, even if you have good ears and good ability to mimic.
You have fallen into a trap which is very common for Americans. It is very wrong to think that there is a single "British accent".
There is no such thing as a "British accent". That is an incorrect assumption. There are hundreds of very different accents in England alone, and they are all so different that some of them are almost like a differnet language.
A significant number of other British people don't speak English at all - there are native Scottish, Welsh and Irish gaelic speakers. It is insulting to exclude these from a "Bristish accent".
If you think that a "British accent" is the one where people speak like Hugh Grant or a "Home Counties Lord" then you will not find many people in London who speak like that. You will probably find it hard to meet someone who speaks like Dick van Dyke in Mary Poppins. London is the least typical of all English cities and the range of accents and languages is enormous.
If you stayed in London long enough you would probably pick up more of the English idiom rather than an accent.
That is, you would still sound like a yank to an English person but you would not say things like "I have gotten" or "be with you momentarily" or "stay on the sidewalk" etc, which are American ways of speaking. Instead you would say "I have" or "I'll be with you shortly", or "stay on the pavement". You would learn native English grammar and words (which is actually very different from the US version) but still keep your US accent.
Having said that, there are a few Americans who have lost their origninal accents completely and now speak with the local English accent of the place where they live. But they are very rare.
You would tend to find that over time, your accent would moderate to become more like the people around you. And the younger you are when you move, the more like the local accent it will eventually become. (Are you 11 - in which case you shouldn't be on this site - or being your mother today?) But it is unlikely to become exactly like that unless you work at it.
And what is a British accent anyway? We have hundreds, so we tend to have a good ear for them and often be able to tell what part of the country someone is from just from listening to them.
A British accent? There are lots, several in London alone. There is no single "British accent".
As you grow older, it becoms less likely that your accent will change naturally. A Canadian relative of mine lived in UK for over 60 years and still had a Canadian accent.
However, you are bound to pick up a bit of it. People in USA would probably say you sounded British.
Yes, in my experience some people very quickly and unconsciously pick up the accent of speakers around them. Others not so quickly.
