Well one thing has become fairly evident from the progress of this thread. The British are proud of and enjoy their diferences from the rest of the world, and don't hesitate to defend them.
And Firia, first of all well done for having the insight to realise that most American portrayals of the British (English in particular) are annoying (to us) stereotypes. I am pleased that you wish to research.
Most of the advise given here is pretty sound and there isn't much more that I can add. Just try to remember that although the British Isles are alot smaller than the USA, the diferences in culture up and down the country are as varied as they are up and down the entire length and breadth of yours. Just ours are alot closer together and changes happen alot quicker. For example, an accent can change entirely by driving 20 or 30 miles away from where you started like Edinburgh and Glasgow, completely diferent versions of the Scottish accent. Then when you take say Newcastle, Manchester and Liverpool, all up in the north of England, all major cities within a couple of hours of each other, yet with completely diferent accents. I feel am rambling a bit and not really helping with your comic in terms of "Don'ts". So I shall depart by just saying: Don't underestimate the diversity of this place, imagine the USA compacted down to our size.