Thanks for all the great advice everyone! My rear dérailleur is an Ultergra medium cage.

I managed to try a spare 105 shifter from a friend and exactly the same problem - so not a shifter problem.

After some trial and error it turned out to be the rear cable housing. The original didn't feel particularly slow, but it's amazing the difference changing that made! I think I will get the hanger aligned for good measure too, but even just changing the housing has resulted in almost perfect shifting.

Thanks again.