Read through Configure Azure Storage firewalls and virtual networks as part of checking and verify the virtual network and firewall rules were configured properly on the storage account. I was able to access storage account file share via UNC path from on-prem network and could connect to storage account via https from App service slot, but not access via UNC path from App service slot. On capturing NSG and App service diagnostic logs and examining the outputs, I could not see any indication of outbound network traffic from the App Service slot to the storage account via the virtual network integration feature, and on reading though App Service networking features, I came across this section and have concluded that this probably applies to outbound connections as well:
I have got round the issue by mounting the storage account file share in the App Service slot (Configuration->Path mappings) and then the files can be accessed via a UNC path to the local path c:\mounts\file share..