How do you change/edit the Apple Watch "emergency SOS" phone numbers?

If you are just trying to edit the phone numbers, you cannot do that from there. You can remove the current emergency contacts and then add new ones. As I said, you edit phone numbers for people in your Contacts. To be honest, if you have phone numbers in your Emergency Contacts that are not in your contacts, I have no idea how they would have gotten there. There is no other place to add or edit Emergency Contacts except in Health where I directed you. If you go to Emergency SOS in Settings and scroll down to your Emergency Contacts listed there, you are also taken to your Medical ID to edit those contacts when you tap on the blue text under them. This is the same for the Watch.

If you are not seeing this, can you explain where you are and what you are seeing when you get to Edit your Medical ID? Does that page not appear? This is the section you should be seeing.

Yes, you go to the Health app. Tap on your icon in the upper right area to bring up your profile. Tap on Medical ID, then tap on Edit in the upper right corner. Scroll down to Emergency Contacts. You get the information from their contact card, so if you need to change a number, change it in the contact card first, then edit the Medical ID.

That information is based on your region, unless you are referring to your emergency contacts. Please explain.

You're welcome. Glad you got it worked out.