I have resolved the issue - and it was none of the suggestions above.
Both my main PC and my laptop have personal and corporate OneDrive accounts on them. The corporate OneDrive worked at home for both, but the main PC never worked with my own account at home... but the laptop did work perfectly with my personal account - but only at work. At soon as it came home the server became "unavailable". It must be my ISP, in this case BT.
I've just been looking at my BT account settings and I have customised Parental Controls running, one of which was File Sharing. I've turned that off and added www.onedrive.live.com to the white list. Bingo! OneDrive is now working for both accounts at home.
Credit where credit is due - this was not a Microsoft error, but one caused by BT's (optional) content filters. To BT's credit, the changes took effect within seconds.