cushioned vinyl is fine in a bathroom. Comfortable underfoot, the rooms are small enough to do it in one piece, and if necessary it can be rolled back to get at the floor.
Unless the floorboards are very flat and smooth, put down 4mm ply first, or the vinyl will go into creases and holes. Put the factory-cut edges of the ply to the middle of the room, because they will be perfectly straight with square corners and will fit together perfectly, and any cut edges go to the edges of the room. After a couple of weeks, when the vinyl has relaxed to size, you can put white silicone sealant in the gap under the skirting, it will look smart, tone with white paint, and prevent water, grime and wildlife getting into, or coming out of, the gap. Clean the gap first with a paintbrush and vac.
It also works in a kitchen, but is more likely to be dented or scarred by heavy furniture or appliances being dragged across it. You can get a non-cushioned very hard-wearing version as used in commercial kitchens, it is more expensive and can have a non-slip stone-like surface.
I suggest asking around for a local flooring and carpet company (not one of the High Street chains) that will do supply and fit. The sort of company that does pubs, restaurants and offices. They may not have a glossy showroom or a glossy salesman.