Ok, it's sinking in slowly.... I did mis-state one thing: the "server" is using one of these, and no AVR Arduino.
So, it appears I can implement the entire functionality using the ESP8266 library. Though it's not obvious to me how to do UDP on this, as there seems to be no UDP class as part of the ESP8266 library.
The clients are using the module I linked to previously, so I will use WiFiESP for those.
Is there a decent web page that explains this well? I've spent a lot time Googling, and just gotten more confused.
Thanks much for your help!
Regards,
Ray L.