The stottie is the headline here, before the ham and before the pease pudding. A stottie, or stotty cake, is a Northeast bread: a large, flat, dense round, soft-crumbed and slightly chewy, cooked hard on the base so it has weight and structure a sliced white loaf does not. In Newcastle the ham and pease pudding combination is read as a stottie first and a filling second, because the bread is what makes it a Geordie sandwich rather than a ham sandwich that happens to contain split peas. The defining fact of this build is structural: a heavy, robust loaf is being asked to carry a heavy, soft filling, and the stottie is the one bread on the British shelf built to do exactly that without surrendering.
The craft is the match between the bread and the load. A stottie is split rather than sliced and is dense enough to take a thick, wet-ish, generous filling that would reduce soft white to paste, which is precisely why this pairing settled on it. Pease pudding, the thick split-pea purée, is spread on in quantity for body and an earthy, mildly sweet counter to the meat; the cooked ham is layered against it salty and cold so the two read together in the same bite. The stottie's chew stands up to both and gives the sandwich something to push back with, so it eats as substantial rather than soft all the way through. No sauce is needed: the pease pudding is the moisture and the body, and the bread is strong enough not to need waterproofing the way a thin slice would.
The variations are the same Newcastle logic with the parts swapped. The same ham and pease pudding on soft sliced bread is the lighter, plainer parent that gives up the bread's identity. A saveloy and pease pudding stottie trades cured ham for a fried red sausage; a ham and stuffing stottie or a corned beef stottie keeps the loaf and changes the filling. Each of those is its own sandwich and deserves its own article rather than being crowded in here.