Let me correct this guide and all speculation once and for all.
First step is to get the tomahawk.
Second is kills in the prison/anywhere (maybe). Once you hear the sound you are set to move on. I'm sure its less than 30 as I usually do this step with the zombies from the laundry plane part step.
Third you go to the bridge and complete a whole round without shooting a shot, killing everything with the tomahawk. Once you hear the sound on completion of a round, you're good. The very earliest that this step can be completed is at the end of Round 10, making 12 the earliest round you can have the Redeemer.
Fourth, go and throw your tomahawk into the fiery pit in the center of the prison. If it isn't accepted you did something wrong.
Final step is on the round after the one where you lose your tomahawk to the pit. Go in afterlife to the place where you got the tomahawk originally and pick it up. Voila, you're done.
These are the exact steps. For the last month and a half I have been playing this map and yeah, this is how it's done.