A bacon and egg bap and a bacon and egg butty are frequently the same filling, and the bap is not an interchangeable word for the bread. A bap is a specific soft roll: round, flour-dusted, with a tender open crumb and almost no crust resistance, built to compress under a thumb rather than tear. That structure is the whole reason the bap gets named here, because the filling it carries is a fried egg with a runny yolk sitting on rendered bacon. A bap closes around that load, presses gently against the egg without splitting it on contact, and absorbs the yolk and the bacon fat into its open crumb as a feature rather than a failure. A crusty roll would resist the bite, shatter, and shoot the egg out the far side. The bap is engineered to lose that fight on purpose.
The build is a moisture-management exercise that the bread is half of. The bacon is cooked until its fat has rendered and the edge has crisped, which gives the soft, wet egg something firm and salty to sit against so the inside of the bap is not uniformly soft. The egg is fried so the white is set but the yolk is still liquid, because the yolk is the sauce: when the bap presses down it breaks and runs into the crumb and the bacon fat, binding the whole thing. Butter on the cut faces is structural, not seasoning. It waterproofs the tender crumb so it holds together through a broken yolk and a fat bleed for the minute or two between the counter and the last bite. The bap is the right size for the load, generous enough to take a rasher or two and a whole egg without the bread-to-filling ratio tipping, which is what lets it be eaten one-handed standing up rather than cut and plated.
The variations are mostly the same sandwich under a different bread word and the breakfast it belongs to. The bacon and egg butty is the identical build named for the regions that say butty instead of bap, the bread shifting from a soft round roll to soft sliced bread or a local barm. The sauce question, brown or red, arrives the moment the egg does and complicates it. Sausage joins or replaces the bacon, and the Northern Irish potato bread and Scottish tattie scone route the same egg through a different fried carrier. Each of those deserves its own article rather than being crowded in here.