I haven't been in the situation where I needed just one penny in either country, but I've had experiences in the US and UK where the store clerk didn't feel like making a mass of change they'd just eat the 1 cent or 1 pence. I suppose it depends on how stringent their accounting is and how much static they'd take from the boss.
As for miserly behavior, I base the vast majority of my English stereotypes on my inlaws. They can be frugal about things, but it's not that they're more frugal than I am, but that the things they choose not to spend money on are different than the things I choose not to spend money on.
For example, they drive old cheap cars, rarely do preventative maintenance on them, and when the car breaks down to the point where it's not economical to fix it, they just get another one. In my family we buy new cars and take really good care of them until they have 100k miles on them, and then start looking for a replacement. Neither way is necessarily better; just different.
So to extrapolate my inlaws to the rest of the population of the UK, maybe it's not that everyone in England is cheap, but that the things they're cheap about are different than what the OP is used to.
p.s. To the OP: you may find people are more open to your opinions if you stay away from declarative sweeping generalizations like "Brits have a tendency to be petty and miserly". It's like saying, "Americans are fat," or "Americans are loud". It's not possible that all 60 million people in the UK are petty and miserly, and the ones who aren't (or people who are married to the ones who arent) might take offense.