Not quite. The Copper Age, also known as the Chalcolithic, is generally considered the end of the Neolithic and the advent of the Bronze Age. It's start date is different in different parts of the Southwest Asia. The earliest evidence that I know of copper smelting is from around 5000-4500 BC, but interestingly from both Serbia and Spain, so two quite geographically separated regions, with Serbia being slightly earlier at around 5000 BC. So what's the earliest weapon? Note that I'm talking about smelting, not the usage of copper through other means. Like most metals, it is possible to cold work chunks of it. The most famous example of such is likely the iron artifacts from Bronze Age Egypt, which came from meteoric iron. The majority of early references to swords that I know come from the late Bronze Age, say the 2nd Millenium BC and onward. For instance this one from Britain which is about 3000 years old. However, the absolute oldest sword I have heard of come from Turkey, and dates to around 3,000 BC. The sword is made from an arsenic-copper alloy, meaning we count it as a copper sword. So for the purposes of this question, that's our start, 5000 years ago. Interestingly, there's a big gap between the earliest known copper items in the early Neolithic around 9000 BC to smelting around 7000 BC and then swords 3000 BC. Kinda cool.

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Anyhow, so when did the first iron or steel sword come into play? See this is tricky, because we don't know to what extent early people separated between iron and steel. Personally I don't believe they did. Yalçın is an expert on a lot of this stuff, and argues that iron began being used in weapons around 1800 BC, and within only a few centuries and the onset of the early Hittite empire iron was considered an every day material. In the referenced text, the author quotes a letter to an Assyrian king (dated to around 1200 BC) where the following line is noted:

...concerning the good iron which you mentioned in your letter...

Yalçın notes that the Hittites may be talking about the quality of iron. However, he also notes that based on archaeological finds from the time period, the Hittites were not only aware of carburised iron (steel), but were using it. So he further suggests that they may be speaking of steel! He further points to a gift to pharaoh Amenophis III (1400 BC) which notes a steel dagger. Personally I think he's convincing. The earliest references to steel that I know of is also from Anatolia and around 1800 BC.

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So, what does all of this mean? Well, the earliest known copper sword that we have evidence for occurred at 3,000 BC, with the earliest steel sword some 1500 years later. And the atom bomb? 1945. So if we count from around 1400 BC, that puts us at just under 3500 years! I mean I'm being a bit loosey-goosey here with the dates. But roughly 1500 years between the first copper sword and the first steel sword. And 3500 years between the first steel sword and the nuclear bomb. But still, 1500 years is a pretty significant period of time between copper and steel.