Sounds like you have an grip shift mechanism on the bike. Personally what I would do in your situation is try to by an old style friction shifter in your local bike shop (if you can still get them) and a new gear cable. Then take the old cable out by undoing it from the derailer and then pulling it out through the old shifter, be careful to note where the cable is routed on the frame as if goes down to the derailer.

Fit the new shifter on your handle bar where your thumb can easily reach it. Run the cable through the shifter and down through the frame (through the cable housing tubes as needed). Put the shifter in the minimum tension position and fix the other end to the cable clamp on the derailler. Pull enough tension that the cable is just taught with the chain on the inside ring. Tighten everything up and you should have functioning gears again.

Hope this helps.