Thank you for the details of the issue you are experiencing with FaceTime. I recommend reviewing the full article below, but the following section specifically stands out to me for the issue you described:
Wi-Fi network issues
- Ensure that your Wi-Fi network is set up using the recommended settings for Wi-Fi routers and access points.
- If you encounter issues using a Wi-Fi network, use standard Wi-Fi network troubleshooting to resolve interference and other issues.
- If you are using router, firewall, or security software that restricts Internet access, verify that the ports documented in Using FaceTime behind a Firewall are open.
- FaceTime requires a broadband connection. Performance may vary according to network throughput and usage.
Thank you for using Apple Support Communities.
Thank you for the additional information. Due to the nature of your issue or question you may find more information by using another one of Apple's support channels - https://getsupport.apple.com/GetproductgroupList.action
All the best,