T W You haven't said which model of regulator you have, but if it is the traditional LUCAS elcctro-mechanical type, Terminal A is where the power comes out. Depending on the battery voltage and electrical demand at the other end of this wire ie the ammeter end, the trick box regulates the power output (regulator section). The unit also isolates the battery from the charging circuit when the engine is stopped (cut out section).
BSA Service Sheet 804 gives a description of operation and test procedure.
Most common fault is dirty connections and poor wiring. Bullet connectors suffer internal corrosion, copper wires oxidise, verdigris and dirt are other enemies. Check the basic wiring system is sound. Checking the regulator for electrical function is easy, but don't fiddle with the adjustment, getting that right is more difficult. The unit needs a good earth connection, try a wire from terminal E to the earthed terminal of the battery.
Your earlier post said the lights stayed the same, so this means there is no power from terminal A to the main loom. Horn is usually wired direct from the battery, so this may be a sign of a component failure or non standard wiring.
Swarfy.
Additional. The earlier posted link to Priory Magnetos gives a circuit diagram to test the regulator. As it stands this test will only confirm the operation of the cut out. In other words with everything connected as shown, the battery voltage should be the same as noted with the battery disconnected, confirming cut out operation and no electrical leakage through the system.
Starting the engine should result in a rise in voltage from terminal A, which will raise the battery voltage as confirmation of some regulator output. As far as I can see this step is omitted. If all is in order, engine running increases meter voltage reading and causes the test bulb to brighten.
Looks to be a final page missing detailing this step. Also there is no test procedure given for the dynamo armature, which would be a useful addition to the otherwise concise text.