Put everything in that Sturmey Archer hub back together, all beautifully cleaned, oiled and regreased. Reinstalled the wheel I'd finally cracked it when I took it out for a long evening ride and it didn't miss a beat. Glorious. Put the bike back in the bike shed feeling very satisfied.
Took the bike out two days later, started riding it and ... it's back to slipping around in second gear around like fish on a waterslide.
Took it all apart again tonight to have a closer look and think I might have found the issue.
In second gear (which is where the gear slips), the clutch engages directly with these teeth in the gear assembly. On mine, the corners of these teeth are worn down. I reckon that's causing the clutch to slip off. I can temporarily solve the problem by tightening the gear chain so that the clutch engages higher up these teeth....but that causes problems with the alignment in the low gear instead.
My gear assembly has only four teeth. But by the look of the SJS spares list, all the replacement gear assemblies have many more, which presumably spreads the load and prevents the teeth wearing so quickly.