I want to buy a 4-channel remote wireless trigger to fire these flashes. Will this work with my Nikon D5200? They work in the camera shoe but will they work wirelessly?
So long as the flash trigger voltages are safe for the triggers, then it should work. Be careful about using these on the camera, as some older vintage flashes can actually fry a camera's hotshoe. Your D5200 can take 0-250V across the hotshoe, and some triggers can take as much as 300V, but some can take far less to fry.
Old vintage flashes have been known to deliver voltages in excess of 350V, so you may want to check out the old Botzilla page either to see what others have measured your models of flash with, or how to measure for yourself.
Also, it's unlikely that an '80s vintage flash will work correctly with your camera/triggers for doing anything but firing in M mode with you manually setting the power level on the flash. TTL, HSS, camera menu control, etc won't really work. Autothyristor modes might if they're available, help you automate power settings.