I have a strange issue that I cannot find others having.

Without WiFi calling enabled on my iPhone 6s Plus, visual VM works fine.

As soon as I enable WiFi calling, notification and upload of new voicemails stops working. Even after I then disable WiFi calling, it doesn't fix itself until I reboot the phone or move out of my current area to another cell!

I spoke to EE and they said they thought it was because the BAIC feature (no idea what that is) was not provisioned on my HSS (home server). This hasn't made any difference.

On EE's website it indicates that visual VM may fallback to normal VM while on WiFi calling. It doesn't, because I don't get the conventional notifications either.

As a result I am not knowing VMs are there, sometimes for a day or more. Anyone else with both these features and having no problems?

I've had ongoing issues with VVM while at home for a while.
Mainly because even if you have WiFi it needs a cellular signal to authenticate.
I get messages coming in 4 hours later sometimes.
I have now turned VVM off as I can now easily get my messages over WC.
Also I've found I get way less VM now anyway.
Not really had issues with WC affecting VVM though.

I think you hit the nail on the head and my association with WiFi calling was just a factor that seemed to be influencing it when it wasn't. It is just unreliable. I have consistent network coverage. It's just that the VVM is fundamentally flawed on EE. Sometimes the notification comes straight away. Sometimes its hours later. And sometimes it doesn't come at all until the next VM comes in, then both come in together. Sometimes a change of cell, rebooting the phone, or coming out of Wifi calling can be a helpful trigger to get the notification.....

I have also noticed if you go to Greeting, that even with 5 bars of 4G coverage, sometimes it says "Failed to fetch outgoing message", and other times it is instant

I have also found that setting Greeting to "Default" doesn't do anything, and it keeps your last recorded greeting anyway.

Not impressed!