The TR4A originally used a special "horn" relay; with one side of the coil internally connected to the same terminal as one of the contacts. That terminal was labeled C2 (according to the diagrams), and should get power through an in-line fuse (not part of the fuse block) directly from the A1 terminal on the control box.
When the horn button is pressed, it should ground terminal W1 of the relay, which should cause the relay contacts to close and connect terminal C1 to terminal C2, supplying 12v to the horns.
If your replacement relay has 4 or more terminals on it (as most modern automotive relays do), you may need to install a jumper between a coil terminal and the hot contact terminal to get it to work. But standard relays don't hold up well in this application, because of the strong inductive kickback from the horns. It's best to use a relay designed for horn service (which should be available from any auto parts store).
If you probe at the relay with a voltmeter or test light, you should find power on 2 of 3 terminals (W1 and C2). When the horn button is pressed, one of the previously hot terminals (W1) should go to ground, while the previously grounded terminal (C1) should go to 12v. If all this happens, then the problem is the wire to the horns, or the horns themselves.
Quick test for horn and power is to temporarily install a jumper (eg clip lead) at the relay from C1 to C2. That should cause the horns to sound, even with a defective relay or horn button/switch. If not, you've got a wiring problem (or again bad horns).
That should be enough to get you started; report back if that doesn't solve the problem.