The default, in the UK, would be a medium mature cheddar. It's strong enough to taste cheesy, but not so strong that you can't eat a slab of it. A few other cheeses would also fit the bill, but they tend not to come in blocks. That is a slab cut from a block of cheese.
We don't normally cut it quite that thick, in a sandwich, but it's not so big as to off put most English. The raw onion would be a lot more divisive.