Thank you Jeremy, that worked! I disconnected and reconnected Jetpack, and then connected a personal google+ page.
For the business page, the prompt asked for offline access (I'd checked that the account that manages the page didn't already have wordpress authorised, but I still got that instead of the "allow this app to share content with your circles" message). I clicked accept anyway, and it seems to be connected - the sharing settings page in wordpress says that it is.
I'll keep an eye on it and make sure that it is receiving the posts - there's one scheduled for tomorrow - and if so I'll resolve this issue. Many thanks for your help!