How do I change the phone number of one of my contacts?

Go to Settings/Emergency SOS. Tap on "Edit Emergency Contacts in Health" at the bottom. Tap "Edit" at the top of the screen.

Alternative: Open the Health app. Tap Medical ID. Tap Edit at the top of the screen.

Where does it stop working? It works on my phone.

Have you already read this support guide?

Use Emergency SOS on your iPhone - Apple Support

If nothing helps, contact Apple Support and explain your problem:

Contact - Official Apple Support

Regards

Giulio