A few things are worth noting: no state just blunders into a nuclear weapon. It requires both the technical capability to do so, and the explicit decision to do it.

West Germany definitely had that capability in the postwar years, and there were serious discussions in the USA about what they would do if it ever tried to "go nuclear." The other European allies in particular didn't want this to happen — Germany had only recently been an enemy. (The Americans called this the "German question" — might they go Nazi again?) And it was for the Soviets a nightmare scenario, so it was seen as something that would ratchet up tensions rather than relieve them.

The West Germans however knew that they would be in the bull's eye of any sort of major European Cold War confrontation, nuclear or not, and wanted reassurances that the US et al. would not see them as purely sacrificial, a territory perhaps to be lost if it preserved a wider peace.

These kinds of forces led to various "confidence-building" measures in Europe include NATO's formation itself, the emplacing of US troops and even (US-controlled) nuclear weapons in Germany, and even contemplating a "Multilateral Force" (MLF) that would allow German commanders to have some autonomy over the use of NATO nuclear weapons. The latter was scrapped (a bridge too far for everyone involved), but the others were undertaken.

Similarly the US also entered into bilateral technical agreements with West Germany to help them with their peaceful nuclear research, in part to make the West Germans have no doubt whose orbit they were in, but also because the philosophy was that if they were going to playing with nuclear reactors then they'd be better off doing so with US assistance, so the US could in effect make sure they weren't doing anything dodgy or going in the wrong directions. EURATOM was a system for doing this in Europe in general.

The long and short of it is, with West Germany the US felt they could and should keep pressure on them to not go nuclear. With the UK they never felt this, both because in the early postwar days they did not fully appreciate the proliferation issue, and also because the UK had been a partner on the Manhattan Project and it was assumed they would go nuclear at some point, especially once they were excluded from further cooperation with the USA. With France, the US was willing to go pretty far to try and discourage them from developing nuclear arms, but France does what France does, and they recognized that if it was dead-set on the notion, they couldn't stop them (they weren't willing to, in essence).

One should not take the West German thing as a forgone conclusion, something inevitable. Even today analysts worry about major nations withdrawing from the NPT and starting their own programs, and there wouldn't be much that could be done to stop them other than diplomatic negotiation or economic sanctions. Both Japan and Australia have had policymakers at times who have flirted with the idea. The Americans considered this a live issue even through the 1960s. The did many active measures to make being non-nuclear a much "easier" option. Such is perhaps the nature of negotiation and political manipulation: you make the option you prefer the path of least resistance.

I don't know much about how they saw Italy. With each of the nuclear states the situations are often quite individual, e.g., with Pakistan you have a very long-term development that the US did know about but felt constrained by its geopolitical needs in that region of the world to let them go ahead and do it, but did subject them to sanctions after they tested. More complicated.

A very comprehensive take on US views on proliferation over the years can be found in Richelson's Spying on the Bomb. On EURATOM and similar measures, John Krige has written a lot on the subject in many articles, and just recently had a book come out: Sharing Knowledge, Shaping Europe: US Technological Collaboration and Nonproliferation.

China and India are "less important" than Italy?

The cost of building a nuclear weapon, much like today, was extremely expensive. The US, UK and Russia could afford to as, at the time, the main cost involved was the knowledge and equipment needed to develop them. Those countries could then develop multiple nukes for a fraction the cost once the first have been developed.

Another reason might have been, with the knowledge that their most powerful Allies had nukes, why would they need them?