I would just check the wires with the battery off, test with a multi meter, as most fault of not charging is the relay, iver in the alternator or on the retern systerm that checks there is a battery connected.
Cut-off relays are 30 years ago, no such thing these days and hasn't been for god knows how long.
@ the OP
Disconnect battery
Disconnect alternator wiring
Check for continuity between thick lead at alternator end, and positive battery lead
Check continuity between alternator body and battery negative lead
If both above are good, reconnect battery, switch ignition to position II (after ensuring those loose alternator wires aren't touching anything or each other), and you should see battery voltage at the end of the thin wire when measured with respect to ground.
If all above tests are positive, you either have a slack drive belt or the alternator is f**ked internally somewhere along the line.