For context I'm an American.
How can I get my American colleagues to stop mocking and imitating my
accent and spelling? Is this something I can/should take up with HR?
I can't help wondering if these guys are being playful but don't know you well enough yet for anything more insightful than the obvious surface variations. If this truly bothers you then by all means it should stop. However, please first be certain that it even should bother you.
Common to every job I've held (warehouses, forklifts, high-security offices, HIPAA-regulated environments, large corporations, tiny mom & pop cottages) is the tendency to tease coworkers, just a little fun--with a slight but seriously sincere undertone. There's more to it than jokes; this immaturity carries a bonding element that's very carnal indeed.
We live a third of our lives with these people at work; shouldn't we all get along and share some fun moments now and then? You wrecked your car this morning? We're all genuinely relieved you're unhurt, but we'll henceforth mock your driving abilities, even if you've never had a wreck before, and even if it wasn't your fault today.
Lacking details my perspective is entirely conjecture, but I'm more inclined to lean toward friendly indeed. This all could be merely evidence of endearment and of acceptance into the fold. I'd likely joke with you in the same way, and I guarantee it wouldn't carry malicious intent. I'd also be disappointed--if not slightly hurt--if you never reciprocated.
While I don't expect cultural differences to be a huge factor, maybe they are. While a thumbs up in America means 'great,' it's a nonverbal expletive in Iran. My friend dated a Russian emigrant who was insulted that my dog of ten years was named 'Sasha.' She piously stated that "we never give human names to animals, it's offensive." Well, I tried to be understanding, but her tone suggested that I was wrong, despite just now learning of this cultural taboo that's only in effect 9,000 miles away.
She'd moved to America and so must accept that she'll meet dogs named Hank and Sally--even if she's not required to like it. With these variances in mind, maybe office joking in Britain is unheard of. However what I've experienced feels more instinctual than cultural.
When we tease it's generally an attempt at deepening a friendship more than a divisive desire to belittle someone.
Of course your assailants' verbal tone could strongly contradict everything I've said; I may be completely wrong but genuinely hope I'm right, and I also hope you come to enjoy the teasing. Because many Americans can be incredibly stubborn, so while you absolutely could involve HR, it wouldn't likely end to your satisfaction.
Please let us know how it goes.