It's not.

Nuclear weapons require a lot of sophisticated machinery to enable their development and manufacture - they require an industrial revolution, which has only happened once.

Second there's a lot of science behind them - there's no evidence that anyone had discovered radioactivity, let alone fission prior to the 19th century.

The nail in the coffin is that they leave an atmospheric signature, which would be detectable in the modern era - and we have detected no such signature.

Pretty much zero credibility.

The building of nuclear weapons depends on so many underlying technologies it's impossible for us to have missed. Refining nuclear material requires large quantities of uranium ore, dangerously concentrated chemicals, and advanced metallurgy. The plants for refinement would be easily comparable in size to ancient temples. They would need to be made of stone or concrete and have miles of piping. The waste material from the plants would be radioactive for hundreds of thousands of years.

There is no way a civilization this advanced could disappear without a trace. Something, ruined cities, old mining sites, garbage dumps, pieces of metal, written records, etc. would have been found.

Whose theory is that exactly?

Aside from it being a technological impossibility (as /u/lindypenguin mentioned), there's nothing in ancient sources* to suggest anything like this ever happened.

*At least the ones I'm familiar with.

Edit: I was curious and googled for it (I shouldn't have). It's the usual lost civilization motif supported by dubiously interpreted events and random quotes from ancient texts. What irks me most is that these people apparently are unable to use proper citation, even when using a direct quote. If you're copying & pasting something it should be easy to also note the book/verse/page number. (Of course they're all copying the quote from each other, instead of going back to the 'source'.)

The people who argue for this do so on the basis of vague scripture describing things being blown up and pillars of smoke and so forth. If we wanted to assume some kind of literal, naturalistic interpretation of these (which is a dubious way to regard ancient scriptures), there are plenty of natural possibilities here — meteoroids, volcanos, really big fires, etc.

To jump from such vague descriptions to "nukes" is just intellectually lazy, if not stupid. Nuclear weapons are a very specific technology that one cannot just stumble into. You might as well as if they had jet aircraft, or lasers. For whatever reason, people who believe in this "ancient wisdom" stuff find it perfectly plausible to imagine that ancients had nuclear technology on the basis of essentially no evidence, but nobody ever supposes that they had a magnetically-levitated monorail. I suspect it results from their own lack of knowledge of how nuclear weapons work and are created.

Well beyond not being credible, the theory is absurd. It is not even worth engaging with, in the same way that someone arguing (based on, say, the Antikythera mechanism) that Plato had access to an ancient version of the Internet would be worth engaging with.