Without makes and model numbers we can't know what connections you have.
In general you would connect the VCR/DVD to a composite video input on your TV for playback. Even if it has HDMI out the VCR won't play through it only the DVD. You could add an HDMI switcher to add additional HDMI inputs,
As to recording it would depend on what outputs you had on the Tivo and Roku and what inputs you have on the VCR.
You could get an matrix HDMI switch with two HDMI outputs. One would go to the TV the other to an HDMI to composite video + audio converter that would connect to the input on the VCR. You could select one HDMI source for the TV and one for the VCR to record.
No way to manage this remotely or even conveniently since you have to set the output of the source to the channel you want to record and set the VCR to the time you want to start and stop recording.
Can't you time shift with the Tivo? Do they have an app for setting it up online?
Anything on the Roku should be available on demand. The quality of the recording on the VCR will not be even good SD let alone HD.