I installed Raspbian on the Pi's SD card, and got the Pi Zero to successfully boot up and connect to my home network via Ethernet (I don't have the Pi Zero W, but instead the model without wifi). I managed to SSH onto the Pi with PuTTY, and when I tried to access the Pi via VNC, the connection just times out. I don't know why I would be able to connect to the Pi via PuTTY and not VNC. Any ideas as to why this might be happening?
It is unclear what you have done to run VNC. It usually doesn't run out of the box. It may be possible that you have to do some installation/configuration to run the VNC-server in particular you are headless on the RasPi.
You already have managed to use ssh to login to the RasPi. That is needed to install/configure VNC from the command line. How to do it please follow the official documentation at VNC (Virtual Network Computing).