There are generally two types of people who develop an accent while being abroad: people who have lived abroad for many years and people who have visited abroad for many days. The latter have invariably just returned from the UK or Australia (never Mexico, fancy that) and insist that they just “picked it up”.
This is going to piss some of you off, but no, they did not just “pick up” that accent. What’s worse, when they returned to the US, they sounded like a bloody idiot. In fact, when I return to the US, I sometimes hear people trying to fake British accents and it’s gut-wrenchingly painful to listen to. Even in the UK I heard Americans trying to fake a British accent and, for some reason I’ve never understood, they were almost invariably young women.
So do I have a “European” accent? Nope. I’ve lived here for years and I still sound solidly American. There is one tiny problem I have to worry about: I say toe-mah-toe instead of toe-may-toe. I say flat instead of apartment. In fact, many of my word choices are distinctly British, but that’s after living there for years and getting tired of people constantly talking about my choice or pronunciation of words. As a defensive measure, I started using their words, though I could never get the accent. My accent has certainly changed in the years I’ve lived here, but it’s not European, whatever that is.
After leaving the UK, I sometimes had trouble being understood in the Netherlands not because their English was poor, but because they spoke American English, not British English. I’ve ordered a white coffee and gotten back a latté. I tried to nick a fag and ... well, you can see where this is going. My friends and colleagues got used to my curious mish-mash of English and American words and that was that. Now in France, no one notices at all what words I use, but when I go back to the US, I better not thank someone with cheers instead of thanks lest I want to come off as damned idiot trying to impress people.
So no, if you move abroad, you likely won’t get the accent. That’s actually OK because many people will adore your local accent. You wouldn’t believe how many compliments I’ve had about my boring, generic mid-west accent. Research on the topic I’ve read indicates that by the time you’re around 20 years old or so (this can vary considerably), your accent is pretty much fixed and it takes a lot of effort and time to get rid of it naturally. Even actors are only deliberately faking it for a short period of time. There are several well-known British actors who use American accents to great effect but fall back to their natural accent when off-camera.
And for that American friend of mine who insisted upon magically having a British accent after two-weeks abroad, suddenly switching to milk in your strawberry-cumquat tea was a dead giveaway.