The fixings were loose, and no amount of tightening made a difference. So I bought a seat fixing kit. The nylon plugs were more secure, and the Allen keyed bolts were easier to tighten than the Phillips-head originals.
However, the fitting on the replacement kits was smaller than the original, and the seat would not clip on. So I used the replacement bolt and plug, with the original fitting, which worked fine. Refitting the seat, I found it more secure, but it still wobbled. So next I added silicone gaskets (cut from a cake pan).
The gaskets made the mounting more secure, but it still slipped. So the final step was to add 340 mm of PVC Pinchweld to the front lip of the seat. This fits around the outside of the front quarter of the toilet bowl, and locks the seat in place, stopping it moving sideways.