I've only been to the UK once, and so my only basis on 'British' peas is 'mushy peas' for the most part. I do know that American varieties of fruits and vegetables are much more likely to be bred for sugar content as compared to European bred ones.
I suspect that most of what you were getting in the UK were 'hull peas', as another name for them is 'English peas'.
As I don't know how peas are classified in the UK (other than 'mange tout'), this is what I know for US peas (may be incomplete; I couldn't find an authoritative exhaustive list):
- snow peas : aka. sugar peas; available fresh or frozen, as a whole pod, almost flat with very little peas inside; you eat the whole thing (mange tout)
- snap peas : aka. sugar snap peas; available fresh in the spring and sometimes late fall. Sold as whole pods with large peas in them (sometimes so large they start flattening each other); you eat the whole thing (mange tout)
- hull peas : aka. garden peas, shell peas, English peas; available fresh in the spring as whole pods, but the pod is not edible. Some varieties are sweeter than others.
- field peas : aka dry peas; Used as a fall cover crop, and commonly sold dried or used as cattle field.
- peanuts : (might not be a pea; definitely is not a nut) : Bends over as it grows, so the pods grow underground. Can be sold fresh ('green peanuts'; highly perishable, only available in areas where it's grown) or dried ('raw peanuts'), but commonly available in the US in a cooked form (mostly roasted, but also fried and boiled (seasonal) in the US south)
And to differentiate by processing:
- fresh peas : it's possible that there are places that sell them already shelled, but around here I can only get them shell on, and can typically only get snow peas out of season.
- frozen peas : typically labeled as 'sweet peas' or 'petite peas'
- dried peas : I'm not sure if these are made from garden peas, field peas, or both
- split peas : dried peas that have had their skins removed and split in half; cook up similar to lentils
- canned peas : foul things; I'm not sure if they're really food; avoid at all cost
And then we have all of things with 'pea' in the name, but are actually different families of plants (I think they're all beans / legumes / pulses):
- pigeon peas
- black eyed peas (cowpeas)
- the 'peas' in 'peas and rice'
- crowder peas
- cream peas
... and if anyone knows of any naming differences between the US and UK, please add it to Translating cooking terms between US / UK / AU / CA / NZ