The prefix "anti" is acceptably pronounced both ways, however it is usually pronounced [antai] (or to a lesser extent [anti]) when stressed or emphasized, and [antɪ] as in 'lid' when said otherwise.
(Native American English speaker here.)
With one possible exception, I don't think I've ever heard an American say “dodgy”, and this includes Americans who deliberately include bits of British English in their speech as an affectation. For example, Americans I've known who say “petrol” don’t say “dodgy”. However, I think most Americans I know personally would understand “dodgy”, especially in context. And if someone wanted to fake a British accent, some might well go out of their way to use the word “dodgy”, though probably not all of them would think to use it.
The one possible exception is an American housemate I had on long trip to Scotland last year. I can't remember for sure if she said “dodgy”, but in six weeks, she had unwittingly picked up some Britishisms, including “got it sorted”. “Dodgy” seems pretty easy to pick up unwittingly.
I certainly do hear “dodgy” from British friends and co-workers all the time. I think I'd use the word with them without it seeming like an affectation.
Other Americans' experiences will be different, of course.
Best Answer
What you are hearing is not mum as in mother, but ma’am, contraction of madam, with a strongly reduced vowel. In British English, it is mostly used as a sign of repect for a woman of superior rank, say, in the military or police. In the film clip, Keeley Hawes appears to be playing the Prime Minister, whose bodyguard addresses her as ‘ma’am’ with the reduced vowel. At the end of this clip from Casino Royale, James Bond (Daniel Craig) addresses M (Judi Dench) in the same fashion.
In British English, the full vowel is reserved for royals:
In American English the vowel is never reduced and may be used as a polite form of addressing any woman, especially one unknown to the speaker:
Some Americans might address younger women as miss in the same context.