This is the question where the honest answer is more cautious than the one most people want, and where ignoring it produces the most spectacular failures.
The principle
Tile adhesive bonds to whatever surface it meets. If that surface is paint, the tiles are attached to the paint, and the assembly is only as strong as the paint's grip on the wall.
Paint is designed to form a thin decorative film. It is not designed to carry the weight of ceramic tiles, adhesive and grout hanging off it. A square metre of tiling weighs a great deal more than a square metre of emulsion was ever engineered to hold.
Test it before deciding
Two quick tests tell you most of what you need to know.
The tape test: press a strip of strong tape firmly onto the painted wall and pull it off sharply. If paint comes away with it, the bond is inadequate and the paint has to go.
The scratch test: score a cross-hatch into the paint with a blade. If the paint flakes away from the cuts rather than staying put, the same conclusion applies.
Also rub the surface with a dark cloth. If it comes away chalky, the paint is degrading and nothing should be bonded to it.
By paint type
Gloss and eggshell are hard and non-porous. Adhesive will not key to them. They must be thoroughly abraded to a dull finish and primed before tiling, or removed.
Vinyl silk is the worst case. The vinyl content makes it both non-porous and slightly flexible, and it is prone to peeling away in sheets under load. Removal is normally the right answer.
Matt emulsion is porous, which sounds promising, but it is also weak and often chalky. It behaves as a bond breaker between adhesive and wall. Light abrasion and a suitable primer can work on sound matt emulsion, but any doubt should send you to removal.
Anything flaking, blistering or applied over damp must come off entirely, along with a diagnosis of why.
What good preparation looks like
Abrade the whole surface to break the film and give a mechanical key. Wash down to remove dust and any residue. Apply a primer appropriate to the substrate and to the adhesive being used.
Then use an adhesive suited to the situation and follow the manufacturer's instructions on the primer, which frequently specify a drying time before tiling.
When to strip back to the substrate
In a bathroom or shower area, strip it. Wet areas are unforgiving, the consequences of failure are water in the structure, and no primed emulsion is worth that risk.
Strip it also where the paint fails either test, where it is flaking, where it is vinyl silk, or where you are using heavy porcelain or stone.
The South London complication
Much of the housing here is Victorian and Edwardian with original plaster that may be a century old and carrying many layers of paint.
Stripping paint from old lime plaster frequently pulls the plaster with it, and what starts as preparation becomes a replastering job. That is not a reason to tile over the paint. It is a reason to have the wall assessed properly before you buy tiles, so that the plastering can be priced, done and allowed to dry first.
A tiler who tells you the wall needs preparing before they will work on it is protecting you from a failure that would otherwise be blamed on them.