I would recommend to try again setting up the chromecast according to this guide in the Stadia helpcenter. There are also some troubleshooting steps in case the setup doesn't work that you might want to try.
Personally I know it can sometimes take some time, but if the Chromecast still doesn't show up in your Home App, you can also try these steps for your phone and the app.
When I first did that I needed to restart my router for example because I had some connection problems with the app and the chromecast, that might also be something you can try.
Hope these steps help you!