This is a newbie talking here. From my viewing experience, British humour is more focused on the visuals and voice lines. American humor also delivers with voice lines, but sound effects and music are added to emphasize the situation.
Just take American Shark Tank vs U.K. Dragons' Den. They are both shows about investing in businesses. However, with Shark Tank, you're going to notice way more sound effects whenever someone mentions spicy information. Cue the dramatic music on top of that too. With Dragons' Den, the camera zooms in and out of facial reactions whenever a witty remark is made.
Gordon Ramsay's Kitchen Nightmares shows these differences too. In an American restaurant, you're going to hear squishy sound effects when he picks up some expired ingredient. We are treated to a fine montage of his angry remarks in the meantime . When he's in the U.K. though, he turns to more of a mentorship role with the people in charge. And of course, clever jokes are made here and there.
Overall, it's about exaggeration for both sides. They both have their own ways to make the situation larger than it really is.