This is an interesting question that I don't have a specific answer to, but can sketch out the boundaries of the answer.

So, did they always have it? No. The early, bomber-based nuclear arsenal had nothing like a two-man rule. It did require the collaboration of many people, though, to effectively arm a plane and use the nukes, so the idea of a single person "hijacking" the operation is pretty remote. We also know that the weapons of the Eisenhower era had no "positive control" mechanisms at all — if you had physical control over the weapons, you could use them. That doesn't preclude something like a "two-man" rule, but we do know that the concern for this kind of safety had not yet fully sunk in yet.

The place for a "two-man" rule is in a missile command — where you get into the arena of "push-button" war. Was the two-man rule there on even very early missile systems? I don't know. That's the question I'd be interested in knowing the answer to. We know that at a certain point — e.g., the deployment the Minuteman systems, early 1960s — that a two-man rule had become standard in such situations. We also know that by the Kennedy administration (1960s), there was an increased emphasis on "positive control." A review of Command and Control systems by the Institute for Defense Analyses in 1975 (Study S-467) notes of the Kennedy years that: "Controversies continued for years over the application of procedural safeguards like the 'two-man rule,' which required several persons at every level of operation of handling of nuclear weapons, or the strict safety regulations in storing, transporting, or otherwise managing them."

But was there a two-man rule in the Atlas system? (It seems like it, from what I've read.) What about for IRBMs, like the Snark or Regulus or Jupiter? I don't know and would be interested in knowing. It doesn't sound like these procedures were always in place. (This timeline suggests that it was only made a formal requirement in 1962, but that doesn't mean it wasn't implemented in some form previously — there were missile systems using this approach prior to this point.) But I haven't seen anything (despite having kicked around the command and control history literature quite a bit lately) that made it quite obvious, yet.