I had this exact same problem with a Canon AE-1 + Speedlite 155A—it would power up and flash via the test button but was never triggered by the camera—and solved it yesterday: by removing the four tiny screws around the hot shoe connector, you can split the case on the Speedlite flash unit. Very quickly, I noticed that, of the several tiny wires that snake into the hot shoe connector, the *yellow* wire was split/broken. I was able to strip about 1/16" of wire on either end of the break, solder it, and then wrap it with a tiny piece of electrical tape. Once I managed to get the sides of the case back together and everything buttoned up, the flash unit began working correctly again and was automatically triggered by the AE-1.

NOTE: it is quite difficult to get the flash unit back together, once the various components have been bumped around. Can be done, obviously, but takes quite a bit of time and lots of patience to reassemble, especially without damaging any wires or connections in the process.