I do understand what the OP was saying, it's more or less just saying a few words in order to not stand out. I don't like having to tell my whole life story to whoever picks up on my American-ness. People always want to know where I live/where I'm from, and I feel like it's rude for me to just ignore them.
So, I normally would just say itty bitty things like "cheers" or "thanks" to avoid conversation when i'm not in the mood. But if I'm ordering something, I try to enunciate better (especially my t's), and when I talk to my DB's 7 year old brother, I have to put on a little bit of an English accent in order for him to understand me.
Also, whenever I return to the US after spending time in England, I realize that my intonation is all sorts of crazy. It causes a good laugh amongst friends, but it sort of comes about naturally for me....I'm not sure if it's just from spending so much time around British people, or subconsciously trying to "fit in" (or, "not stand out") while out in public in the UK.
But I will NEVER try to feign an English accent amongst my British friends...that would just be wrong!