Hi Alasdair,
Welcome to the world of lilypond vertical and horizontal spacing, a perennial topic of varying degrees of difficulty.
You may do well top refer to the NR Section 4.4.2 Explicit staff and system positioning. There you will find you can specify the exact positioning for staves. If you do that. you most likely need to place explicit \break statements in your score to lay out staves as you want them.
But there's more than one way to do it. You may also consider NR Section 4.1.4 Flexible vertical spacing \paper variables. There you will find you can adjust basic-distance, minimum-distance, padding, and stretchability for systems. Sitting down with a good coffee or possibly something a little stronger you can do a lot with this set.
Alternatively, you can also have lilypond remove empty staves as desired. This may not be appropriate for you instrumental score, but it is also worth bearing in mind.
Andrew