This is something I really struggle with internally, probably more stress than most people should have about it!
I never want to lose my American roots and how I speak, but I find it is easier to say certain words differently here and then I don't have to explain what I mean or get teased. It really does make you self conscious all of the time. I am constantly thinking, will people think I am pronouncing the word "wrong" or will they know that it is said differently in the US? (words that come to mind are: process, schedule, herbs, garage, route)
I work as a Teaching Assistant and I definitely have to use the British words for things otherwise they will look at me with a blank stare.
I do think that the Americans in the UK probably have it easier that the Brits in the US, where there is not a big popularity for British TV shows or movies.
I definitely switch the words I use when I am speaking to British people or Americans and I change my spellings as well. I sometimes feel like I am fluent in another language!