The butty in a bacon and egg butty is the bread, and butty almost always means soft sliced bread folded over, not a roll. That is the distinction worth leading with: where the bap is a round roll engineered to compress, the butty is two flat slices of soft white that lie flush against the filling along their entire face. Flatness changes how the sandwich behaves. A fried egg with a runny yolk laid on rendered bacon between two slices is pressed evenly across its whole width rather than gripped at a single round point, so the yolk breaks in a controlled spread instead of squirting from one side, and the fat is distributed into a broad area of crumb rather than pooling in the base of a roll. The word is regional, the bread is the substance, and the substance is a flat soft carrier doing a different job from a bap.
The build is a yolk problem solved by timing and by the bread choice. The bacon is cooked until the fat has rendered and the edge firmed, giving the soft egg a salty, structural counter so the fold is not all softness against softness. The egg is fried for a set white and a liquid yolk, because the yolk folded into bacon fat is the entire sauce and a hard yolk leaves the butty dry. Butter spread to the edges of both slices seals the crumb so a broken yolk and a fat bleed do not go straight through the bread before it is eaten, which is a real constraint on a sandwich assembled in under a minute and eaten with one hand. The fold matters as much as the slices: a butty pressed flat holds the egg and bacon in a single even layer, where a roll lets them shift, so the slices are chosen soft enough to yield to that press without tearing. It is built to be quick, generous, and stable rather than tidy.
The variations are the same filling under the other bread words and the breakfast around it. The bacon and egg bap is the identical idea named for a soft round roll instead of folded slices, and a barm or a cob is the same again in other regions. The sauce divide, brown against red, lands as soon as the egg does. Sausage joins or replaces the bacon, and the Northern Irish potato bread and Scottish tattie scone carry the same egg on a fried base instead of bread. Each of those deserves its own article rather than being crowded in here.