The reluctance (read: outright refusal) of Sturmey hubs to shift under load came as something as a shock to me, having previously only ridden dérailleurs and the occasional Shimano hub gear. Until you get a feel for it, you pretty much have to stop pedalling - simply easing off as you would when changing down on a modern dérailleur system isn't enough. Static force is a problem too - it'll be just as reluctant to shift if you're holding the pedal against a brake or the weight of the bike uphill.
I defer to the others on the specifics of the 3-speed hub, but will make the general observation that Bromptons are a fairly harsh environment for cables, and if the indicator chain thingumy is correctly adjusted, the cable (especially if it's old or of unknown providence) would be next on my list of things to check.
The Brompton shifters have a reputation for being pleasingly reliable, in spite of their plasticy appearance.