You won't normally get any publishing, but if the artist wants to encourage your version
and thinks there's something in it for them you might get a cut. Everything is negotiable
(assuming the artist owns their publishing of course).
You might own the
recording, but the whole thing might then work the other way. If you're the one getting
the benefit, the artist might (keep his publishing and) ask you for a cut of the
Who's on top in this deal? They will get to choose...