So, I see that there are only a few ways to sever someone's astral cord: Astral Dreadnaught, Gith'yanki Knights or higher have a silver sword. The last option is where my question lies.  Psychic winds can cut the cord too, but I can't find the answer anywhere. Does it have hp and the psychic damage hits this directly? Otherwise I don't see how to play this out.

There are no rules for this; it's probably old text from previous editions that wasn't updated properly.

So I'm having trouble finding out how the Pyschic Winds can sever the Silvery Tether while in Astral Projection. I get how the Gith'yanki have Silver Greatswords that can do it on a crit, just like an Astral Dreadnaught can but under the description for Psychic Winds it doesn't show an option for cord cutting. Please enlighten me so I can scare the crap out my players. :)

Sad unfortunately. :( I may have to assign a number on the chart.

It might get addressed if they ever come out with a planescape setting for 5e. You can check older edition rules.

It is probably meant to be a save that if failed by a certain amount causes it.

Maybe it's a "if the wind reduces the target to 0hp, it severs the cord" sort of thing?