@DEustace , I’m not an Apple guy, so can’t speak on the specifics of the Apple watch. I’ll offer some thoughts in general.
How are you connecting your Apple watch to Zwift? Are you using Bluetooth? If so, I believe you can only send a BTLE signal to one receiving device. BT is a one-to-one communication method. If you already have your watch BT paired to your bike computer, it won’t talk with Zwift also. When you look in the Zwift pairing screen, do you see the watch pairing with Zwift and showing your heart rate there before starting the ride with it already connected to your bike computer?
For the Garmin HRM, same questions. Are you using BT or ANT+. If BT you should have the same problem. If it is a Dual mode HRM, then you could use ANT+ to talk to both Zwift (if you have an ANT+ dongle in your computer) and your bike computer as ANT+ allows multiple devices to listen in on the same transmission from the HRM.