“Britain's very first action after declaring war on Germany in World War I was to have the cable ship Alert (not the CS Telconia as frequently reported)[12] cut the five cables linking Germany with France, Spain and the Azores, and through them, North America.”
I believe they had enough wireless stations to get across the Atlantic or they could communicate via diplomatic bag through neutral countries/ones with intact cables
Also, how did the Zimmerman telegraph happen if Germany to Mexico was cut off?
The telegram was sent to the German embassy in the United States for re-transmission to Eckardt in Mexico.
https://en.wikipedia.org/wiki/Zimmermann_Telegram