Short answer: at least as early as the Eisenhower administration, NATO commanders had some 'predelegated' command authority for the use nuclear weapons in certain extreme circumstances — generally, a major conventional war with the USSR in Europe for tactical nuclear weapons, or the incapacitation of the National Command Authority for the strategic arsenal.

The long answer is: it's complicated, because tactical nuclear weapons would only have been used in extreme circumstances that would have consequences far beyond the battlefield. The US maintains to this day a tactical nuclear arsenal, but they've been out of fashion for decades because they're so utterly impractical — they're not a serious part of anyone's nuclear strategy anymore. Rather, they're 'prestige' weapons.

Eric Schlosser wrote an excellent book last year on the development of the US nuclear weapons programme, and particularly the evolution of nuclear weapons safeguards and command/control policy: Command and Control: Nuclear Weapons, the Damascus Incident, and the Illusion of Safety.

A recurring theme in the book is the complexity of the inter- and intra-service rivalries, and the machiavellian politics of nuclear control. At the top level, you had a rivalry (which still exists today) between the US Navy and USAF for control of the strategic deterrent, but you also had rivalries within the Air Force over how US nuclear doctrine should look, and under whose operational control it should fall.

The tl;dr version is that, eventually, the Strategic Air Command won most of those battles, and "became America's preeminent military organisation, its mission considered essential to national security, its commander reporting directly to the Joint Chiefs of Staff." The Navy retained a pivotal role in the nuclear deterrent, but the dominant, prestigious command was SAC — and the bureaucratic/chain-of-command faultlines persisted. It wasn't until 1992, when STRATCOM was formed, that nuclear weapons were centralised in one unified command.

Tactical/battlefield nuclear weapons were always an awkward, subordinate part of the US nuclear doctrine, which pretty quickly arrived at a doctrine based on a massive, hopefully decisive first strike as the only way to 'win' a nuclear war.

Politically, battlefield nuclear weapons were primarily the preserve of the US Army, and so they lobbied hard to defend their arsenal. Tactical nuclear weapons were problematic to use for any number of reasons — the Davy Crockett recoilless rifle, for example (a nuclear bazooka, essentially) had a maximum range "so short—about a mile and a half—that the soldiers who fired it stood a good chance of being killed by it."

But more pressingly, the use of tactical nuclear weapons created a colossal risk of escalation: their use represented an escalation from conventional to nuclear warfare, which would likely very quickly move from tactical to strategic; from the European battlefield to the cities of the continental USA and USSR. The sole expected use case was a massive Soviet invasion of western Europe — and even then, nuclear artillery was generally regarded as a last-resort response to overwhelming force; as a way to slow the advance and to create room for manoeuvre. That's why at various times NATO commanders on the ground had command authority for the use of tactical nuclear weapons pre-delegated to them, to be exercised in exigent circumstances.

But that battlefield doctrine was generally accompanied by the caveat that a conventional war in Europe was likely to turn into a strategic nuclear war anyway — and that process would only be accelerated by the deployment of tactical nukes.

Edit: I'm slightly rethinking that statement that tactical nuclear weapons are "not a serious part of anyone's nuclear strategy anymore." That's not entirely true — India and Pakistan both maintain tactical nuclear arsenals that do form a significant part of their deterrence strategy. That's in large part why the prospect of confrontation between India and Pakistan is so unsettling. What I really meant to say was: tactical nuclear weapons are out of favour among the major nuclear powers (US, UK, Russia, China, France), which all favour strategic deterrence.