What's the difference between peas in the UK and in the US?
I've lived on both sides of the Atlantic and I've never noticed much difference in peas, but then I'm not a connoisseur. The UK generally has 2 kids of peas on sale on the store, regular peas (aka garden peas) and petit pois, which are smaller, younger peas. The petit pois are much sweeter than garden peas, which are bigger and less flavorful. I've never seen petit pois in the states as far as I can remember, but there are generally peas and "sweet peas", and sweet peas tend to be sweeter than regular peas. The UK also has a product called mushy peas where the peas are mashed up and served as more of a liquid - not as bad as it sounds, it can be tasty if done right. This would often be out of a can if served at home. 

As for how to get your peas how you like them I'd ask whomever makes dinner in your family what type of peas are used and how long they are cooked. People in the UK used to boil vegetables until they practically disintegrated but cuisine has improved dramatically in the past decade and nowadays it's much rarer to get that.