You can read all about it:

https://en.wikipedia.org/wiki/Nucular

But it's not something to be horrified by; it's no different from how we commonly pronounce "February" without the first "r", or "government" without the "n", or "Wednesday" flipping the "dne" to "nd".

What's even more interesting is that it's only in the context of weapons/energy. The same person will say "nuclear family" the way it's spelled.

But in the weapons/energy context, it's just a natural re-use of the suffix in "moleCULAR", "oCULAR", "cirCULAR". Technically wrong in terms of its derivation, but it feels entirely natural to say, and requires less tongue movement.

It's not due to a lack of education or anything. It's more like a regional dialect, where the region is nuclear weapons and energy.

Do you pronounce it as Wed-nes-day?

I've never heard it as anything but Wends-day, but maybe everyone else is wrong.