After 1.5 years of active development, I believe EIP-1559 is finally ready to be included in a network upgrade and would like to propose it be included in the upcomming London upgrade.
The work on EIP-1559 was done in two "phases", the first during 2019 and the second from ~March 2020 to now.
The first phase was mostly driven by Vulcanize, which put togehter an implementation study as well as an initial implementation, and came on AllCoreDevs to discuss the EIP. A few technical objections were brought up regarding the EIP, mainly the risk of DoS attacks, but it was nevertheless moved to "Considered for Inclusion".
Around March 2020, the Besu team re-started the work on EIP-1559 in earnest with the goal of addressing the objections raised on AllCoreDevs and from others in the community. This work has been tracked here.
While there are still things to iron out in the client implementation, I believe that there are no "major issues" outstanding on the EIP and that it is ready to go through the normal network upgrade testing process.
Finally, while the EIP does give an overview of its benefits, I've written a more extensive description of them here.