Sturmey archer 3 speed problem

Cheers for that mate. The majority of that makes sense but there's always plenty of scope for me to fuck these sort of things up. I'm useless with instructions - the better half can vouch for that. I'll give it a try when I get in after work. No doubt I'll be back here. Possibly with pictures for you to kindly annotate. :)

lol @ 'gives it the beans'. Ha! That's exactly what's happening.

OK,
the 'ickle chain' is called the indicator chain, though what it's supposed to indicate is anyone's guess.
It disappears into the hub on a threaded rod which should be tightened up all the way, then backed off half a turn to allow the chain to go in the right position.

Still at the back of the bike, the ickle chain is connected to a narrow barrel adjuster: a thread and a locknut, to fine-tune the cable tension.

The next point of contact is the fulcrum clip. This is where the bare cable enters the plastic sleeve. On older bikes this is a metal band which can be loosened and slid up and down the chain stay to give you coarse tension adjustment. On modern kit there is a fixed length arm bolted on to the dropout, a proper cable clamp, and a bigger barrel adjuster, as you'd find on brakes.

What's happening with your better half's bike is that the tension is too slack and it's enough to slip the gear momentarily into neutral when she gives it the beans.

If there is no tension at all (the loose chain) then it will be in 3rd gear. Really you want the chain to be just under the slightest tension in 3rd. This should make 2nd and 1st engage perfectly and not slip. Use the big adjuster at the fulcrum clip to get it near enough, and then fine-tune it with the narrow barrel adjuster

Sturmey's official instructions say adjust it in 2nd, but initially doing it in 3rd is easier, and you can fine-tune it in 2nd.