but all the places where configs and connections should be stored are empty on my device.

The configs for wifi networks are stored in:

/userdata/system-data/etc/NetworkManager/system-connections

I think they are here because /userdata won't be overwritten with ota update

Ofono configs:

/userdata/system-data/var/lib/ofono

There are no configs for gsm network in the NetworkManager directory. I think, this is because gsm networks are read-only in NM

https://www.redhat.com/sysadmin/becoming-friends-networkmanager

@adeliepenguin Thanks, very nice tutorial

I wonder how the image for OTA updates is being made. Are there any docs on this? I used to run a LTSP server with disk-less clients and also made some custom live DVDs, so I'm not afraid of the read-only root file system. But configuring either of those was fairly easy, since in the end, it was just a directory tree with a nearly vanilla Linux, plus some clever scripts (pre-written) to wrap it up. I suppose there are more stages required for UbuntuTouch, cross-compiling at the very least.
What about generating one's own image of UbuntuTouch? I know it would involve a high risk of bricking my device, so I'd probably not do it, but I'm really curious here.

$ sudo

?

[nmcli ...]
lists the connection as read-only (mobile data / gsm). Maybe there's a nmcli command to make it writable. I didn't find one so far.

(The following also applies to mobile data only, not WLAN.)

In case anyone is interested: The command would be

nmcli c m  f651d7ab-521b-e3e9-f9fa-721e2b64ec45 connection.read-only no

But it does not have the desired effect. The output is "Error: failed to modify connection.read-only: the property can't be changed."

From the nmcli docu:

Key Name: read-only
Value Type: boolean
Default Value: FALSE
Value Description: FALSE if the connection can be modified using the provided settings service's D-Bus interface with the right privileges, or TRUE if the connection is read-only and cannot be modified.

I've done some reading, and I'm quite sure now the read-only property was set intentionally. I found lots of tutorials, but all the places where configs and connections should be stored are empty on my device. (I haven't used wifi so far, so that left no traces, either.) Since a phone has a built-in gsm connection, I suppose this connection has been pre-defined with a default config in UT, not expecting that anybody would want to modify it. (Just upgraded to RC66 today, no difference.)

From what I learned, it's really network-manager who is in charge here. It has a plug-in to talk to ofono for modem use, and eventually delegates DNS to dnsmasq (local DNS service). Setting a custom DNS server would be done exactly with the commands @Capsia posted - if this was allowed.

BTW, I especially liked this tutorial (albeit it did not help me in the end):
https://www.redhat.com/sysadmin/becoming-friends-networkmanager

I gues this is not a lack of permissions, but a feature

This is probably because the gsm connection isn't managed by the NetworkManager.
I think it is managed by ofono, but I might be wrong.

nmcli -p con show f651d7ab-521b-e3e9-f9fa-721e2b64ec45

lists the connection as read-only (mobile data / gsm). Maybe there's a nmcli command to make it writable. I didn't find one so far. (I'm desperately missing man pages in UbuntuTouch; didn't manage to install them in libertine, either.)

Just guessing: maybe the root file system has to write-enabled to save this?

I've tried to do it with root in rw and I've got the same results.
Also tried using sudo.

The commands I've used:

nmcli c modify 628f4269-3a24-7858-e4b0-3765d485ecda ipv4.dns "208.67.222.222, 208.67.220.220"
nmcli c modify 628f4269-3a24-7858-e4b0-3765d485ecda ipv4.ignore-auto-dns yes

Get network ID with:

nmcli c

UT: changed DNS for wireless successfully but unable to change it in gsm mobile network

The following error occurred when attempting to save connection settings:

Just guessing: maybe the root file system has to write-enabled to save this?

The results:
Desktop was able to set DNS for ethernet and wireless network correctly
UT: changed DNS for wireless successfully but unable to change it in gsm mobile network

The following error occurred when attempting to save connection settings:

Failed to save '/222071071843368/context1' (628f4269-3a24-7858-e4b0-3765d485ecda) connection: Connection is read-only

BTW - I'd opt for:
CloudFlare:

1.1.1.1, 1.0.0.1
2606:4700:4700::1111 
2606:4700:4700::1001