Educate me, how do they work?

It's just that whenever I 'operate' the accelerator the attachment to the pump doesn't seem to do anything?

hi sam, does the accelerator pump arm move outways when u operate the thottle? i assume it does
this should squirt neat fuel into the carb from the bridge
you can see it if you look down the intake and operate the throttle arm on the carb
but the float bowl has to have fuel in it of course
if theres no fuel squirting, then the diafram in the pumps split,quite common

you can actually hear the fuel squirt in and feel a slight resistance on the throttle/ butterfly arm if all is ok

What is it that's 'not doing anything', Sam?

It all looks to be in order - does the lever on the actual pump move along with the butterfly link arm?

Is the throttle butterfly closed in these pictures? Apart from what ayjay says, the only other thing I can think of is whether the connecting rod is adjusted a bit too short? Ie: in the 'resting' position shown, with the throttle butterfly closed, the pump lever might already be sitting part-way through it's travel, so when the throttle is opened up the pump doesn't have much more effective further travel to go - it's only giving half-squirts...

in the first pic you can actually see the butterfly is about half open