You go back to every airline you ever worked for, beg for your flight logs, hope they are willing to provide them, and then over the course of weeks re-enter th

You go back to every airline you ever worked for, beg for your flight logs, hope they are willing to provide them, and then over the course of weeks re-enter them into your paper logbook or software. Then you hope that the airline will also be willing to sign-off your logbook. TL;DR: you're screwed.