[Arran]

If it is shellac there are french polishing supplies available online if you want to try them, but it's a slow process. I think part of why it has alligatored is someone may have had a go at brushing shellac, or lacquer on top of the original finish, then again it could partially be wax buildup. In all honesty a proper refinish won't devalue the radio, most of these 1920s sets are not worth big dollars where such things matter, so stripping and refinishing the cabinet is an option, if done properly. I would rub the finish down with turpentine, or mineral spirits with a rag, see how it comes out.
Regards
Arran