Originals were never wood. It is an aftermarket. Is there a center button? You said it is an MG wheel so I assume there is an MG button in the center which is probably the original button/logo. Does it push, move or otherwise?
One other possibilty is a fuse might be blown.
You said you have two horns. Look at both. Does one have two electric terminals and the other just one? The horn with two should have a wire going to it and then it jumps (using the other terminal) to the 2nd horn. It is possible the wires are not correct or to the wrong place or missing a wire. This is a common horn wire set up but not sure, being English, how a 1964 MGB would be. I assume the horns are simply grounded by being bolted to the chassis or body. Make sure grounds are good. Unscrew bolts and clean contact to body. If the car is still positive ground then touching a wire to the horn terminal from the negative side of the battery should make a sound. Just brush it quickly against the terminal. That at least proves the horns are OK.