Can I put the Emma Original on a slatted bed frame, or does it need a solid base?

A slatted frame is fine, and it is what most people put it on. The one thing to check is the gap between slats: keep it under about 7cm, because a 25cm all-foam mattress will slowly mould into wider gaps and you will feel ridges within a year. If your frame has wide gaps, a sheet of hardboard or a set of extra slats fixes it. It also works on a divan or a solid platform; on a solid base it sleeps a touch warmer because air cannot move underneath.