Weird pseudo-American accent? 'Lishen Shweethard-let'sh get this strait'

Great movies, but I didn't understand why it was that during the 10-year flashback moments to when they first met, Mark/Finney spoke in a half-American accent that bordered on an extreme parody of 30's Gangster/Scarface-Al Capone/Suits vs. Mugs type-films (i.e., "Eh? See here, copper, youse no right to be heerh").

I first noticed it in its most pronounced form when the two were walking along the road in the scene right before Mark loses his passport and waves down a car to take him back to the hostel where the girls stayed, after which Hepburn pulls his passport out of a bag; he tells her:
"Listen sweetheart, let's get this straight. I've absolutely no intention of
staying with you; I don't know what your plans are, but I have a schedule. I am
NOT, whatever you may be, on holiday...I'm here strictly for the buildings" --
at first, I thought he was being facetious or just innocently joking with a
funny accent."

It's hard to transliterate, but it sounds like he's got his lips pursed extremely tightly, or talking out of the side of his mouth.

I understand that he supposedly has just been in the U.S. for 2 years and returned to Europe some time not long ago. Still, his accent seems more or less British in the later scenes (especially the "present" ones, i.e., as they're flying out w/ their car).

Admittedly, his accent seems strange throughout the movie, where it's not a fully classical British one, which one would expect it to be in a 60's UK film. Nevertheless, there is that weird pursed-lips sort of talking where it's as if he truly is attempting to speak with an American accent in all of the scenes from the period when they first meet and 'romance' with one another.

What's going on? Am I just naive to Finney's regular accent (meaning, does he always tend to speak like this? Is he known for having an accent that fluctuates in films?)? Or is this a nod at the 2 years he spent in Chicago? Something else that I'm missing?

Hope someone has at least a tiny bit of input they could contribute! Thanks in advance.