Israeli scientists and officials considered themselves a nuclear-armed power in 1966, and they most likely had crude nuclear weapons ready to use in 1967 (very crude, apparently — not deliverable from an aircraft, but something that would have to be emplaced by a helicopter, or something like that). They definitely had a usable arsenal by the 1970s, and today are estimated to have an arsenal of around 80 warheads. There is no real doubt on this topic — there is a lot of documentation, including photos of their nuclear weapons mockups taken by a whistleblower. The US government requires people who work for it to say that it is "alleged," but that is a fig leaf, part of a policy that is well-documented to avoid official acknowledgment of what everyone knows is true. Declassified CIA documents confirm that US intelligence agencies concluded decades ago that "Israel... has produced and stockpiled a small number of fission weapons".

How did they acquire them? They started doing research into nuclear technology in the early 1950s. The Israeli Atomic Energy Commission was created in 1952, focused initially on peaceful research, but with an aim to building up the technical base required to make weapons. In 1956, the French agreed to help Israel develop a nuclear reactor capability, knowing that Israel was interested in using it to breed plutonium for bombs. They developed a secret nuclear facility in the Negev desert, Dimona, which was basically a nuclear reactor and, underneath the reactor, covert facilities for extracting plutonium and fabricating it. This was operational by the early 1960s. By 1966 they had conducted some kind of zero-yield experiment that made the scientists confident that they were a nuclear power, despite having not really tested a bomb at full scale.

There are, of course, more details to the story, but that is a very brief overview. There is no doubt that the Israelis have nuclear weapons, though they will not confirm or deny that fact in a deliberate policy known as "nuclear opacity." The idea is to have the benefit of a deterrent (because everyone knows they have nukes) but not fall into the sanctions regimes that would come with being an unauthorized (by the Nuclear Non-Proliferation Treaty) nuclear power.

The work of Avner Cohen is the primary source on this topic — Israel and the Bomb and The Worst Kept Secret. There are many other dimensions I have not touched on, like how knowledge of the Israeli bomb became public, how the US and Israelis came to an agreement about how to treat Israel's nuclear capabilities, and the putative nuclear test in 1979 that Israel may have done with South African assistance. The National Security Archive has some great Electronic Briefing Books on the Israeli nuclear program, with great documents and commentary. Cohen and Bill Burr of the National Security Archive (both friends and colleagues) very recently published an article at POLITICO describing new research into how the US government investigated the controversial 1979 "Vela incident," as an aside.