I have personally only repaired and custom made 532nm mounts, not 473 but there is not too much difference if only replacing the pump.
The fact that this is c-mount is a good start. Use a fine scalpel type xacto blade, and make fine markings where you can on the LD mount. The idea is to mark for reference only, putting in the exact same place only works if you are re installing the original LD. (Though you may get lucky)
Remove the LD with minimal disruption, and don't touch anything you don't have to. This includes the thermal paste on the face of the LD mount and the position of optics.
REmove the LD and check for any additional optic such as a FAC and replace with the same type LD (with or without the fac..).
Once you have mounted the new LD, pay careful attention and try to sit it in the same spot (via markings) as the original.
If your pump is 1W, you may want to turn down the current to the LD just a bit and start at lower powers. You NEED 808nm goggles from here forward.
Power up the unit with the LD fairly firm in place, but not tight.. You want to be able to nudge the LD's corners and work it into the sweet spot.
You have to find a spot where the laser will emit some light for reference, so this may include playing with the current to the LD and minor adjustments of the LD's position at the same time.
From here you will need a power meter. Get your slight output on the meter, and begin moving the LD swiveling from one side to the other in a very slow and accurate sweep. If your mount does not use the center hole of the LD and is a clamp type, you may have to shift its position up and down and side to side. I cant see by the pictures how yours is...
The goal is an obvious one from here, get the highest output! <big surprise..
Once you have found it, slowly increase the current to the LD, back to where it was in the first place. Going beyond can be costly and burn the medium etc.
Total time should take no more that an hour, 2 at tops, just take your time if this is the first time out
good luck
*EDIT:
You may run into mode shifting when you turn the current back up... Should this happen, shut it down, and loosen the LD just a bit.. Turn the laser back on, and try fine tuning the position until you see the beam as desired and the output close to where it originally was. Because you have to loosen the LD, and will be doing the fine tuning at full current, be sure to give the laser a rest when needed and watch it doesn't get too hot.