The cheese salad sandwich is a moisture-management problem with cheese in it, and that is the honest way to describe what makes it succeed or fail. The build is Cheddar plus the standing British salad set: lettuce, tomato, cucumber, sometimes onion or beetroot, on buttered bread. Every one of those vegetables carries water, and water is the enemy of bread. The cheese is the only part of the filling that does not weep, and so it ends up doing double duty as both the savoury anchor and part of the defence. Whether the sandwich is fresh and crisp or a damp grey disappointment is decided almost entirely by how the wet elements are handled, not by the cheese.
The craft is keeping the salad from drowning the loaf. Tomato is the worst offender and is salted and drained, or deseeded, or kept off the bread entirely behind a slab of cheese; cucumber is sliced and laid dry rather than straight from a wet board; lettuce is washed and then properly dried, because wet leaves are wet bread within minutes. The cheese is cut as a thick continuous layer and positioned as an internal wall between the wettest vegetables and at least one face of the crumb, so it is structural as well as flavour-bearing. Butter to the edges on both slices is the real waterproofing, and it carries the cheese's salt across an otherwise lean, watery filling so the whole thing reads as seasoned rather than damp and bland. The bread is soft and plain because the build has no single loud note that wants arguing with, and it wants assembling close to eating, since a salad sandwich is on a clock that no amount of butter fully stops.
The variations are mostly which vegetables are in and which dressing arrives. A version with salad cream or mayonnaise turns the same set creamy and seasons the leaves directly; cheese and tomato strips it back to the single wettest pairing; cheese and onion drops the soft salad for one assertive raw bite. Each deserves its own article rather than being crowded in here.