Not every British sandwich is named for its filling. A long shelf of them is named for the bread, and the bread is the point. The stottie of the Northeast, the soda farl and potato bread of Northern Ireland, the Staffordshire and Derbyshire oatcakes, the Kentish huffkin: each is a distinct regional loaf with its own crumb and crust, and the sandwich made on it reads as that region's before it reads as anything else.
The craft is matching the bread to the load. A dense, chewy stottie can carry a heavy, wet filling that would destroy soft white; a soda farl is soft and quick-staling so it is filled and eaten the same day; an oatcake is closer to a thin pancake folded round its filling than a slice. The format decisions carry the same weight: a doorstop is cut thick on purpose to take a generous filling, a triple or meal-deal pack is engineered to survive hours in a chiller, a panini is pressed so the bread crisps and the filling fuses.
The variations are the bread aisle itself: baguette, ciabatta, focaccia, sourdough, rye, bagel, croissant, and the British rolls each change what the sandwich can hold. Each deserves its own article rather than being crowded in here.