The factory used a contact cement. They'd apply some to the veneer and some to the wood. Well that seems like pulling it all off and starting over. Here's a better idea:

Use Elmer's Furniture glue. Keep a damp wrag around to wipe the excess. You’ll have to occasionally rinse out glue and then wring it out again. Take putty knife and stick it into to void. You may find that the void looks bigger than you originally thought. So, ponder how far you want to go. Notice photos showing how to clamp over larger area. Squirt in some Elmer's and spread it around as best you can with with the putty knife.

Then take some pieces of wood and some small clamps and clamp the lifted area flat with the wood pieces being used between the clamps and base.

Work the glue in. Start clamping. Best to use little pieces of wood on both sides to prevent clamp marks. Squeeze them in place. Let sit a few minutes. Then using a damp rag, wipe off glue that squeezes out. Tighten then repeat till you are not getting more. Leave for a few hours and you'll probably find more to wipe out. Then leave over night.

You're going to have to figure a lot out by intuition. It really is doable.