I think it depends how quickly they can manufacture a nuke. Fuel grade material isn't weapon grade material. I'd say that to be a nuclear power, you'd need to be ready with warhead designs, ready with some subsystems or components already manufactured, ready to process fuel into weapons, and ready with the industrial plant to finish manufacturing and assembly on a turnkey basis.
"We don't have any bombs right now, but we can easily build one in less than a month, and six a month thereafter" - you're a nuclear power.
"We could, in theory, build out a nuclear weapons manufacturing plant, start processing fuel into weapons-grade material, and get to the point of building a weapon within a year" - you're not a nuclear power. But you could in theory become one if you wanted to.