lgtm

One nitpick: It changes the synchronous/asynchronous behaviour. Specifically, it only uses the Logs module which return unit instead of the Logs_lwt module which returns unit Lwt.t. In some cases, this is in line with the previous use, in other cases it's not.
See #609

One issue: In bin/cohttp_server_lwt.ml you do not set a reporter. According to Logs' documentation:

If you are writing an application you must remember to set the reporter before any logging operation takes place otherwise no messages will be reported.

This has been superseded by #609, already merged. I am going to close. Feel free to reopen it if I am wrong