The case ID is 00593600. This was occurring on both of our VeeamONE servers (two datacentres). I managed to fix the issue by;
- Reinstalling (upgrading) 7.0 R1. For some reason the upgrade wizard said the Web component was still on v6.0.
- Re-ran the patch to 7.0 R2.
After this the database magically started working! Prior to this, after doing a bit of Googling it seems I was supposed to run a utility under c:\Program Files\Common Files\Veeam\Veeam ONE Settings to check the database connection. I checked but couldn't for the life of me find the utility, called VeeamOneSettings.exe. The folder was there but no exe or files were contained within it. Strangely, after re-running both the upgrade to 7.0 and the patch to R2, the utility has now mysteriously appeared.