So you have no tools to work with but you do have good common sense.
Let me tell you of some old mechanic tricks. Red Neck micrometer - If you at least have a good adjustable wrench ( cresent ) if you will, you can get real close to the diameter by adjusting the wrench snug to the threads and then without moving the wrench jaw insert feeler gauges until they fit snug within the cresent wrench jaws. Add the total of the gauges and you have the diameter in thousandths.
At the hardware store ( Ace ) find a screw that diameter and lay your threads on top of the screw threads. If they have the same pitch ( threads / inch ) the two will interlock and match perfectly. To make a Red Neck tap of that screw, file a v groove in the screw at about a 30 degree angle at the end effecting the first 3 0r 4 threads. Remove any burrs from filing and there you have a thread chasing tap to correct your screw up.