I skimmed the OP and misread the requirements, I had thought you wanted the DNS to apply globally and had completely missed that you are looking for it to appy to the mobile connection only.
For a single connection only…
nmcli con mod <connection-name> ipv4.ignore-auto-dns yes ipv4.dns "1.1.1.0 1.1.1.1"
then reload the connection…
nmcli con up <connection-name>
<connection-name> should be double checked, on the Librem 5 I have here it is “Handsets” while looking at the screengrab you have provided yours is “NXTGENPHONE”
so for your connection the actual terminal commands would be…
nmcli con mod NXTGENPHONE ipv4.ignore-auto-dns yes ipv4.dns "1.1.1.0 1.1.1.1"
followed by…
nmcli con up NXTGENPHONE
EDIT: the IP address 1.1.1.0 does not seem to be a valid DNS server which is likely to cause issues, cloadflare’s main DNS servers are 1.1.1.1 and 1.0.0.1