I realised that I hadn’t had any Windows Update notifications for a couple of weeks. I got suspicious, started digging and was surprised to find that the WU service was not running, nor was the related BITS service.
I started them manually, checked they were set to auto-start and rebooted. I checked they were both running and ran Windows Update.
After downloading some updates and rebooting, I tried enabling the Outlook Avast plugin, (fully expecting the crash) and found it just started working as expected.
I have no idea what changed but the updates I downloaded are:
2017-12 Security Monthly Quality Rollup for Windows 7 for x64-based Systems (KB4054518)
Update for Microsoft Excel 2010 (KB4011617) 32-Bit Edition
Security Update for Microsoft Word 2010 (KB4011614) 32-Bit Edition
Windows Malicious Software Removal Tool x64 - December 2017 (KB890830)