Best case: the name of the channel (e.g., WRCDT) doesn't change. In this case TiVo should process it as a channel reassignment. They'll be a big message on your TiVo's identifying all the reassignments. All your One Passes will continue to work. All recordings happen as scheduled.
Not so good case: the name of the channel (e.g., WRCDT) does change (e.g., to WRC) . In this case TiVo should process it as a channel delete and add. They'll be a big message on your TiVo's identifying all the deletes and adds. All your One Passes will need to be updated to reference the new channel name. Some recordings missed if you don't edit the One Pass in time.
Worst case: some channel names change, others don't. Changes happen on different days with new names added on a different day than the old names are deleted. Some recordings missed. Generally a big mess that you'll have to sort out manually.
TV networks go by lots of names. For example, the name of its national or local affiliation (NBCUniversal), the name of the station (NBC4), the callsign of a station (WRC-TV), UHF or VHF frequency of the station and its provider-assigned Virtual Channel (4).
Cox is responsible to assign the Virtual Channel (4) because they're amassing all these networks onto their cable boxes. If you press "4" on a Cox remote, the Cox cable-box will tune WRC-TV.
The names of affiliations, stations, callsigns and frequencies are not changing...only the Cox-assigned Virtual Channels are changing.
Virtual Channels really depend of the local affiliates within different markets. For example, when I lived in Maine, CBS was Virtual Channel 5. Here's, it's 9.