How can paper be glued to cardboard without creating wrinkles?
Use SprayMount, or similar - it is formulated for exactly this purpose: fixing paper to cardboard.

What you do is place your artwork face down on a piece of newspaper, spray a single, thin coat on the back, then place it onto the cardboard. It dries almost instantly, so there is no wrinkling, while still giving very good, invisible adhesion. There are some varieties which are repositionable/removable too!