I see that flat spot where the half cock notch used to be. Do not use pliers but use a vise. There is a special tool (mainspring vise), but a bench vise is the easiest. Lock jaw pliers can scratch up things so I'm against using them. Even a 1" C-clamp is better.
If you buy a new tumbler, you must fit it.
Got a vise? Put the mainspring into the smooth jaw vise and compress it. This will allow you to separate the mainspring from the lock.
Remove the sear screw and then remove the sear spring and then lift off the sear.
Remove any bridle screw and lift off the bridle.
Remove the hammer screw.
I use a piece of square brass that is ground to find the hole where the hammer screw was.
Place the lock plate on two pieces of wood such that it supports the plate as close as possible to both sides of the tumbler.
Place the square brass drift and tap lightly with a mallet. No heavy whacks. You don't want to bend the lock plate.
This should complete the disassembly.
Check the new tumbler. It should fit flush against the lockplate. Any canting will cause friction and drag. Smoke and file.
Place the hammer onto the tumbler. You may have to put it in the vise and squeeze it to slip the hammer over the tumbler.
Install the tumbler screw and tighten.
Install the sear and then the sear screw.
Install the sear spring. Be sure to hook it into the notch in the lockplate so that there is tension on the sear.
Install the bridle and screws.
Put the mainspring into a vise and compress it. Place the lockplate onto it such that the curved arm of the mainspring hooks onto the tumbler.
Open the vise.
Now that it's reassembled, test the half cock. Pull the hammer back and it should catch now. Pull the hammer back all the way and it should catch on full cock. Push the back of the hammer slightly. You're seeing if the tumbler is held by the sear. OK, if it does, then put your thumb on the hammer and release the sear. Allow the hammer to move forward. Install the lock. Put a piece of rubber on the nipple and cock the hammer and release it. It should snap forward sharply. If so, then....
Voila! You're done and you didn't know you were an armorer, did you?