You're right to think that one can sound more American. But that isn't the whole story. Both are possible in Br English, and neither sounds 'more American', when the context is appropriate:
[regular 9-5.00 worker, being asked in the late afternoon/evening]
Did you go to work today? [That is, the time is appropriate for me to ask whether you conformed to your regular schedule]
[shift worker, or just someone who habitually works at different times on different days]
Have you been to work today [or is this one of the days when you go in late]?
That's all I've time for now. I'll try to think whether there's a general rule; but this is the way I see this specific example.