What kind of glue to use for **large area** paper to wood that is visible while wet?
This sounds like quite a similar situation to applying wallpaper. Wallpaper is normally hung with wallpaper adhesive, a thickish paste with a relatively long drying time.

The advantage of this is that you can apply it to the paper and let it soak in for a few minutes before attaching the paper to the plywood, which should help ensure you don't miss any bits.