I would sand that crap off and see whats under it.
My choice might be chemical removal. I'm not sure how thick the veneers are, and sanding could cut right through if you get carried away. Use a chemical stripper for the hard work, then sand with fine grits to clean things up. Maybe use some bleach if there are water stains, and then use chemical (alkaline) and dye/pigment stains for adding color.
This summer I picked up a nice ca. 1935 GE console radio. It has some lifted veneers, but they are really thick, surprisingly so. That will make it easy to reglue them. More recently veneers have become thinner and thinner until some are paper thin.
Chris Campbell