If you use the liquify filter instead, you can save your mesh.
Or, here’s a workaround if you want to use Warp:
Convert your layer to a smart object before you warp it. When you want to use the same mesh on another thing, duplicate the warped smart object, and go to Layer>Smart Object>Replace Contents to swap the other thing in (you’ll need to save the thing on its own file first.)
I just tried it and it worked fine.