I am using Brume 2 in 'Drop-in gateway' mode. After swapping the SIM card in the 4G Router to the EE network, Brume 2 stated: "The interface is connected, but the Internet can't be accessed". And it was correct, it really did not.
Finally I found that on activation EE APN was automatically created and it enables both IPv4&IPv6 in its profile. And Double NAT is not possible that way.
Solution is to create a new APN in the 4G router and set the IP Type (or Protocol) to IPv4 only.
I found this doc page but it doesn't seem to go into detail why this message may happen -- the above should be listed as one (possible) reason for that, in my opinion.
(And yes, I know double NAT is not optimal and to avoid that, currently no devices are connected to the LAN port of the Brum since my setup on the server side does not require to send local data through this GL.Inet router.)