Humanity has never had enough weapons to destroy the planet, not even today. An interesting thread in AskScience earlier demonstrated that if the global niclear stockpile were denotated in Antartica, only a very tiny fraction of the ice sheet would melt. Nukes are powerful, but not that powerful.

Humanity has enough weapons to destroy a significant portion urban infrastructure. Dont get me wrong, this would be an absolutely hellish experience to live though. The immediate survivors would have to deal with an unprecedented level of famine and disease, even if they survived the immediate exchange. Most of the population would probably perish. But outside of us, the Earth will continue on, just like normal.

I'm not quite sure what time it was that humans first where capable of blowing up the planet. But I think some context to the state of nuclear arms during WWII might give a preliminary answer.

No, the US was not capable of reducing the earth to dust in 1945. In fact, of the two bombs that were dropped on Hiroshima and Nagasaki (plus the trinity test bomb), they were the only three bombs we had made so far. I'll double check, but I do believe that bomb assembly team at Tinian were anticipating the delivery of enough fissil material for a fourth bomb, but the war stopped before that date arrived. I do know that there was no intention to withdraw the assembly team before the end of the war (which wasn't necessarily anticipated immediately after the dropping of that atomic bombs).

No, I do not think the U.S. held back in the use the atomic bomb in WWII. In fact, the bombing of Nagasaki proceeded ahead of schedule out of concern that the skies would be too cloudy on the 11th. And the almost untouched cities of Hiroshima and Nagasaki were selected primarily because they were large cities that would give good "data" on the power the atomic bomb. And, like I said, the intention was to use a fourth bomb when it became available. Not only that, because the war was hardly over yet, some consideration had been given to the use of the bomb during the planned November invasion. A rather hopeful consideration was given to use nine atomic bombs to clear the beachheads at the three landing sites for American troops (which is really troubling to think about, considering how we now know how dangerous it would be to send people through a mushroom clouded battlefield). Michael Gordin has a great book, Five Days in August that gives a detailed history of the final steps the U.S. took in utilizing the atomic bombs against the Japanese in 1945.

So, in regards to your question, I think on the face of it the answer is simply answered by finding out how many megatons of blast is necessary to blow of the planet, and then looking into the proliferation record for the approximate date that that amount was achieved. I'd hazard a guess that it wasn't before 1949/50, when both American bomb production was relatively slow and the Soviets had not yet built their own bomb. The NRDC has this table showing the number of warheads per year per country (I've seen this table in a few other places, I'm not sure if this is the NRDC's work for from some other estimate). You can see that after 1950, the number of warheads skyrockets. I doubt that the pre-1950s number, though high, were really in a state to be used seeing as the American program for managing atomic weapons before the 50s (I think) was a bit of a mess. The table does not show the megatons of weapon power, just warheads, which might be a little misleading. As the cold war heated up (haha), atomic research was producing more powerful bombs, and the eventual production of H-bombs dramatically increased weapon yield.

But to get at the spirit of your question, that seems to be not so much the calculations for apocalypse but rather, in a general sense, simply at what point did we have enough bombs that their use would make life not so hot for humans. I have a book here, with an essay by Carl Sagan of all people, where he estimates that (in 1983), there were over 13,000 megatons of nuclear weapons in combined U.S. and Soviet arsenals. Enough, he says, to obliterate a million Hiroshimas except that there aren't a million Hiroshimas in the world (at the time, only 3,000 cities over 100,000). So, we definitely know that we had well over the amount needed to killer well over most people on earth by 1983.