I'm not picking up too much of an accent here, really, since I'm 33 now, but when I was 15, I moved from Texas to Chicago for a year and came back with the Texas accent mostly gone and a few funny "yankee O's" in it's place.
I then lost most of that Chicago accent after being in Texas again for so long, but never gained back much of the Texas accent I originally lost, so I sound much more neutral now, whereas I must've sounded like the world's biggest hick when I moved to Chicago.
I do think being younger and more impressionable makes a big difference in picking up local accents. I didn't try to gain that accent, but I did work on not saying things like "y'all" and then it kind of went from there.
I do notice that certain things I say now have a bit of Geordie accent in them, rather than proper Queen's English, since that is how my husband and everyone else around here speaks. I notice I sometimes say "summit" instead of something and certain other things like that, plus some phrases and slang I have picked up. But overall, I still have a strong American accent. I doubt that will ever change now. But that's ok with me.
I'm sure your little ones will have British accents since they are so young. It's hard to believe there is anything cuter than tiny people with English accents. It makes me smile every time my stepson says something very English-sounding (or Geordie... he's taken to saying "Awww Man" in a proper Geordie accent when he thinks something is really funny.
)