One of the most common desoldering methods is to use a desoldering wick/braid (a flux-coated, loosely braided copper rope) in conjunction with a soldering iron: [link]
It can help to frequently cut off saturated parts of the wick, so you have a fresh tip. Doing so with a handheld wire cutter (wire stripper, flush cutter, etc.) is inconvenient, because you have to put down the soldering iron to pick it up.
Instead, why not use a device that sits on your desktop? Just insert the wick as far as it's saturated with solder and press the big lever in front of the opening with whatever part of your hand or arm is convenient. A blade comes down and chops off the used part of the wick, and it falls into a container. Then you can go back to desoldering.
This would actually be pretty easy to build at home, and could even be based on a handheld cutter.
N/A [2019-01-04]-- notexactly, Jan 05 2019 Video showing desoldering using solder wick https://www.youtube...watch?v=htrcZuK_ZsYMentioned in idea body [notexactly, Jan 05 2019] The other idea Less annoying desoldering wickMentioned in my anno [notexactly, Jan 05 2019] I came up with this immediately after posting my other idea [link], as I was watching the video I've linked on both ideas, to make sure it didn't have anything nasty in it. It might have been smarter to check that before linking it on the other idea, but if I did that, who knows if I would have had this idea?-- notexactly, Jan 05 2019 random, halfbakery