I've always been really confused by this aswell. Obviously their mother is American and their dad says she and him met in NY but the film strongly suggests they were brought up in the UK. Its confusing.
"What you don't like rice? Tell me Michael, how can a billion Chinese people be wrong?"
Definitely had to rewatch it again, but I think I got the gist.
The Dad (forgot his name) is Kat's stepdad, and Bunny is both Kat and Amy's mother (making them half sisters). Daddy mentioned to Nick that he met Bunny in New York, where "this little monster" appeared, and suddenly he was a father. I took this to mean that he met Bunny when she had Kat, and Kat was around 2-4 years old at the time. So I assume that Kat was definitely born in America and raised there until at least 4 years old (I would consider really young children to be viewed as "little monsters", terrible two's and whatnot).
So if we assume this information is correct, Kat developed her verbal and speaking communication skills around American accents, hence why she has one. If Daddy and Bunny chose to remain in New York after their marriage and when she had Amy, then it's also possible that Amy also grew up listening and learning how to speak being influenced by primarily American accents.
There is also the scene at Amy's bachelorette party. Amy says that Kat was voted "Best Expat with best eyes, face etc". So again I interpret that scene to mean that yes, they did grow up in the UK for a little bit, but if you are voted "Best" that would imply a high school graduation thing in a year book. Also the "Best EXPAT" thing adds more too it.
So my guess is that both Kat and Amy were born and raised in America, and then in their high school years moved to the UK and lived there for for a few years, met their respective partners, and when Kat's engagement failed she returned back to the USA to work.
Those early years of when you are learning to talk really do influence your speech and talking manner. If they lived in the USA until they were 12-13 or so when they started high school, then it's entirely possible that they had American accents all their lives. And even if they did start to develop an English accent, they would have quickly lost it if they ever returned to America as adults (which we know Kat definitely did).