Tivo is wired only, but the network connection is just to allow devices on the LAN to connect with the TV Anywhere app to control the Tivo (e.g. Change channels using virtual remote on mobile device). Remotely accessing of most functions (e.g. Managing recordings or series links, etc.) would be done via the built in modem and the VM cables so the Ethernet connection is not required.
Disclaimer - I don't work for Virgin Media. I'm just another VM user trying to help out so my answers may be wrong :-) If you do like my answer please mark it as helpful as it may help others My setup: Vivid 200 Optical fibre with Superhub 2 in modem mode connected to a Netgear R7000 router. Telewest/VM user since 2001.