My understanding is that the wire is really that long. 10-20 miles of wire depending on the torp and generally coincident with the max range of the torp.
In a real sub, I assume that the "Torp has acquired the target" refers to the range gate of the torp's active sonar. Once it goes to max ping rate, it's probably safe to assume it has found a target. Not sure if the range gate info comes back up the wire or if the sonar tech monitoring the torp hears it. As others mentioned, they might also hear the torpedo speed up for its attack sprint.
Mike
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