I would mix up lacquer thinner and denatured alcohol....2/3 alcohol and 1/3 lacquer thinner...swab on...let sit 30 seconds and wipe off with paper towels....repeat until all is even....then brush on a finish of your choice.
Linda C

Use Elmer's carpenter wood glue to glue the loose area down & lay a clean rag on it & a heavy book, for a few seconds & then wipe any glue that is around edges & test that way a time or 2 & then use books (like 3-4 inches thick) on it overnight. A lot of other glues will loosen when you go to stain it & put your top coats on, then you have a mess. Repair the loose wood first before using Linda's mix to get old finish off. All the finish is off when there are no "shiny spots" anywhere.

Yes....fix the wood before you work on finish....
I don't lay a rag on the glued spot...I use waxed paper...less likely to stick and if it does water dissolvs it fairly easily.
Linda C

Many of those old radios used a thin veneer for the outer layer of the case, so use caution and be gentle with the wood. It will be easy to sand right through the outer layer.

Clean off the old finish and repair the wood before refinishing. Before destroying the finish, take photos of the details, notable any pin striping and decals around the knobs. You may wish to duplicate the trim decoration.

Thanks everyone for all of your helpful advice. I have discovered that the wood that makes up the top hinged cover is made of some sort of Masonite so I'm not sure how I am going to deal with this. I might just have to go cheap and roll on some contact paper as I'm not sure what else I can do.