Thanks for all the replies 
OK, I’d assumed that going into settings and checking was enough to validate that the microphone permission had been granted. It seems I was wrong.
I’ve managed to get most of the apps to work after various actions. A couple of apps started working when I toggled the microphone permission in the settings, while one app needed to have its storage reset before it re-requested the microphone permission and then worked.
What this means is that each app has been checking to see if it has the relevant permission, and the operating system has replied that it has, whilst somewhere internal to the OS the permission hasn’t been properly granted.
I’m pretty sure all the apps that didn’t work have been installed since I first commissioned the device, and I’m beginning to suspect that some permissions issue has been fixed in one of the OS updates since.
The only app I’ve not been able to fix is GBoard (google keyboard), which is pre-installed in the ROM and doesn’t seem to be fixed by either revoking/toggling permission settings or deleting the storage.
Just for a bit of background: I’m an Android developer (and have been for 14 years).