We have a quite poorly Fukien tea tree that I'm presuming has a fungal infection. It's dropping many of its leaves, which have black spots / are totally black. We've tried spraying with copper sulphate solution but I know that's not a cure. What's the best approach—is it salvageable?

That looks more like an aphid infestation, or possibly wooly aphid, check under the leaves for insects, if you find any then you need to spray with a WEAK soap solution, it looks very bad, it might be ok.

Thanks—yes, there are some what look like greenfly, not white woolly though. Will try soap spray. Thanks!

So, after a few sprays over a couple of weeks, the leaves have stopped dropping, and no sign of infestation any more. Some new growth is appearing which I take as a good sign, but there are still brown marks / spots on even the newest leaves / flowers. Is this still something to worry about, or is it likely to fade?

Marks on existens leafs will never dissapear. Your problem is solved when new leafs are fine.