OK the plot thickens and I have more clues, but understand what's going on even less. Here are the steps I'm taking to give me this clue (if anyone happens to have their Sherlock Holmes hat on)
1) I open the project, and without starting playback, I play a few notes on various soft synths. The latency is there.
2) I then delete an instrument track that has an instance of Battery on it. There is then NO audible latency when I play soft synths.
3) I start playback. Immediately the latency appears.
4) I stop playback. The latency remains.
In fact from the point at which I begin playback, the latency remains whether the project is playing or whether it's stopped, until I close the project and open it again, in which case steps 1-4 apply again.
I really don't understand. Deleting this instance of Battery without playing anything back removed the latency, but as soon as playback starts the latency remains until I close the project and reload it.
Any ideas?