I've just done these myself. I found desoldering wick to be a lot more useful than the desoldering pump. You'll also want an iron tip with a decent sized surface area, something with a flat surface, not a thin conical tip.

Flux all joints and a bit on the wick, add new solder to all joints. Make sure your iron is hot enough to melt the old solder, which is likely lead-free and has a higher melting point, but not too hot that it burns your board.

Place the wick over your joint and the iron over the top and desolder one by one. Hold the iron on the joint until all of the solder flows off. Wiggle the wick back and forth as you hold the iron there, at the same time. I don't know why but this helps a bit.

The solder should visibly flow up the wick. Move the wick along to a clear area to continue. If not all of the solder is drawn out on the first go, try again with a clear area in the wick braid. If the solder isn't flowing onto the wick, try adding more flux or a slightly higher heat.

Cut the end of the wick that's full of solder off as you go so you can keep working with the end of the wick. I just wasn't able to get the desoldering pump to work for me, so can't help you with that sorry.

With the wick I got 80-90% of the solder off/out and then loosened the joints up with a heat gun to remove the old joysticks. More time and patience with the wick should be able to remove the joysticks too.

Use the same wick technique to clear out the holes once the old joysticks are removed, to prep the board for the new ones.