Old self bows most often had oil finishes and that could allow drying over the years. I have collected self bows for years and, in general, I don't consider shooting them a good policy.
However, there used to be a product called Wood Rejuvenator by Wood Medic and it was wonderful. I was successful making quite a few old lemonwood, osage, and hickory bows shoot by repeatedly warming the bow and the liquid, wiping it on in thick coats, and letting it soak in for days. Then I would string the bow with a long string, at about a 3-5" brace height, oil it again and let it sit for a day or so. Then, increase the brace height to maybe 12-13", and let it sit again. Then to about 24", and then to 28". Only then did I draw the bow. I am confident a good percentage of those bows would have broken if I just threw on a string and jerked it back, but I never lost one with this process. Of course, I put on a french polish or oil finish after the bow was working ok.
Rekindling my collecting interest after a 5 year layoff, last year went looking for some more of that stuff. Wood Medic is no more and googling Wood Rejuvenator brought up some new materials, but not that magic stuff that worked so well.
I also used it on laminated bows that showed little finish left and it may have helped....but I couldn't be sure....tho any oil applied to dried old wood surely makes it look better! I think your Cub is likely ok to shoot anyway. Apply some English Oil or some other good wood treatment, maybe with some warmth, and give it a go. I've had a bunch of Cubs from the 50's and always end up letting them go to a new archer or youth who needs an inexpensive, reliable longbow. Have never had one break in my hands or theirs.
I am looking for an old Cub for my wife now, one of the early ones with honey-colored glass, something around 40# or less. Wouldn't you know, I can't find one when I'm trying!