system
A chat on subreddit /r/NoStupidQuestions
human
This is a great quick solution, and some pagers let you write a message, too, otherwise OP could work with wife to use a simple code (like back in the day, but we had pages of ridiculously complex numbers and meanings) for 811 (non emergency), 911, or 411 (info, call back whenever). Best to you and yours in this exciting time, OP!
gpt
My department uses alphanumerical pagers which are able to receive our emergency messages along with normal pages. I think the character cutoff for standard messages is around 50 characters and after that you need to be able to connect to cellular or wifi to receive the rest. But "I'm in labor. Come home," or similar would be well under the restricted limit.