I agree that hot hide glue is the best glue for player pianos.  If
you insist on using Elmer's white glue for veneer work then try this:

Put the glue on the veneer and also put the glue on the part the veneer
is going to be glued.  Now let both pieces dry a little so they are
just no longer tacky.  Use a hot iron to bond the two pieces together.
I used this process on a Wurlitzer 800 juke box that has now been
subjected to Louisiana heat and humidity without any failure of the
joint.  I also spray a mist of water on the outside of the veneer to
keep it from curling.

This technique was in an issue of Fine Woodworking magazine that has
also had excellent articles on different glues.  They have written
favorable articles about hot hide glue on several occasions over the
past years.

Don Teach
1610 E. Bert Kouns
Shreveport,  LA  71105
318 798 6000