How do I stop my phone from saying “an update is required to use cellular data on this iPhone”?

Perhaps it is a "carrier update" you need, are you seeing a prompt which says something like that? If so proceed with the update, which is a small piece of code issued by your cellular provider.

Update your carrier settings on your iPhone or iPad - Apple Support

If you have done everything in this article If you see an Update Failed or Apple Pay Not Available alert after iOS update or restart - Apple Support and it is still not working you probably have a hardware issue.

If you see an Update Failed or Apple Pay Not Available alert after iOS update or restart - Apple Support

For the third time in a few days, my phone is saying "An update is required to use cellular data on this iPhone. But the ipone itself says it is up to date, and I went through an elaborate set of instructions from Apple just a couple of days ago; those efforts supposedly update both the software and the firmware. I suspect something is wrong with my phone's hardware.

Do the required update.

What required update? The one my phone says is not needed or the one I just finished ?

It's not an Apple update you need.

Update your cellular softare by following the link on your display.