How to fix iMessage and FaceTime not working after macOS upgrade on both Mac and iPhone?

I installed the latest Mac OS on my pro laptop based on auto update policy. iMessages, and FaceTime now no longer work. Today y iPhone has the same **** issue. Apple really bolloxed this is "upgrade!" Fix it!

[Re-Titled by Moderator]

Original Title: Latest Mac OS upgrade causes iMessages and FaceTime to no longer work on both my Mac and now iPhone!