Apart from checking again the article that you came to the community from
Use Walkie-Talkie on your Apple Watch - Apple Support
you could check that you have no security apps or VPN that may interfere with the handshake.
Force restarting both watches and phones may help.
Its a bit of a pain unpairing the watches from the phones and setting up again but that is the final remedy.
If you've completed all that troubleshooting and the issue is still not resolved, use the link in the article to Contact Support.
Click here > Contact - Official Apple Support
or phone numbers
Click here > https://support.apple.com/en-us/HT201232