I am not sure what manual changes you made yourselve, but /etc/resolv.conf should be a symlink to /var/run/connman/resolv.conf. And it should point to 127.0.0.1, the connman DNS proxy. Then Connman knows what DNS to use. This works for VPN and non VPN situations I can confirm.
Then I do not know where these nameservers come from.
You can try to use a temporary fix. After connecting via wireguard VPN use
echo 'nameserver 88.252.172.57' >>/run/connman/resolv.conf
or even 1.1.1.1 or whatever DNS public server.
echo 'nameserver 1.1.1.1' >>/run/connman/resolv.conf
Is that a permanent change?
Not sure if it is required or the opposite but as I wrote it is a temporary fix. /run/connman/resolv.conf is/should be overwritten every network change or connman restart.