[Radioroslyn]

Wow that's kinda of a mess. I've had a few cabs that have had some wrinkles near the bottom maybe within the last 4" or so. Like yours could get them flatter but still where pretty noticeable after adding some finish to the job. Gloss really makes them stick out.

I think I would be inclined to remove the veneer and see how bad the underlying wood is. I don't think you'll be able to get what you have to flatten out like it was 80 yrs ago. The fibers in the wood have swollen from exposure to water and they aren't not going back to the original size. If the underlying wood is ok you could sand it a bit, clean out the groove in the front panel to fit the veneer in, and use a few wire nails to secure the back edge of the veneer after gluing of course.  You would also want to remove the base trim on both sides. It's held on with hide glue & wire nails. Removing this will give you access to the bottom 3/4" of the veneer on both sides.

This is just my take on it. I'm no expert just a hobbyist.