Existing quotes will not automatically change. The preference setting is for when entering new ones only.
Also, the type of quotes you get depend in the language that is applied to the text in which you are entering them.
If you type a quote, does it come out curly now? Is the applied language set to English (presumably)? If yes, do a find-and-change from a quote to itself to change all of them at once - replacing them counts as "being newly inserted".
I've quitted InDesign, re-booted, gone into prefs, selected typographer's quotes (it was a default selection), made sure the dictionary is set to English UK, then created a fresh document. It still won't give me proper quotes. I can get by this by going into Glyphs and pasting them in from there, but it's not a great way to do it. Thanks for responding so quickly.
... made sure the dictionary is set to English UK, ...
That is slightly suspicious wording, so I am not yet convinced it is an InDesign problem. Yes: 'the' dictionary has a setting for kinds of quotes. However, it has nothing to do with the text in your document! Which is something you need to check: if you type some text, what language is assigned to it by default? Look in the Type panel, not in the Preferences.