I just tried this on mine. I had never used the Emegency Information stuff yet. I first tried adding it using Phosh Mobile Settings and saw the same behavior as you: saving the changes and returning to the page showed the field empty. I then created the phosh directory:
mkdir ~/.config/phosh
Then created and edited ~/.config/phosh/EmergencyInfo.keyfile with contents:
[Info]
OwnerName=Famous Jameous
DateOfBirth=Some Time before now
And checked the lock screen again and I can see the information I expect. I then tried modifying the settings in Phosh Mobile Settings and that now works as well.
@ASwyD2 Does that match what you did?