They would all come crashing back to Earth, as they have nowhere near enough fuel to even reach orbit around the Earth, let alone transfer to an orbit which hits the sun. Doing that actually requires a huge amount of energy and a big spacecraft.
But let's say we took all the nuclear missiles and mounted them on spacecraft capable of reaching the sun, then launched them? Ok, then what? Just crash them into the sun? Sure, the sun would vapourise them and the gases and plasma they became would merge with the outer layers of the sun. We wouldn't even see anything from Earth. Boring!
What I think you're getting at is what if we detonated them, right? Nothing would happen.
The world's entire nuclear arsenal adds up to about 3 Gigatons of smash. For the scientifically minded that's 1.25x10^19 joules. In one second the sun produces 3.8×10^26 joules.
What does that mean? It means that the world's entire nuclear arsenal is equivalent to one ten millionth of a second of the sun's power output. Or put another way, we the sun would have to absorb a variation of one ten millionth extra energy for that second, then it would be back to business as usual. From Earth you wouldn't even notice the tiny momentary increase in light from the sun.
Obviously the actual physics are going to be more complex than this kind of napkin-sized estimate, but I'm pretty sure the sun would absorb that tiny variation without blinking. If you exploded every nuke in one spot you might disrupt the sun's plasma enough to make a noticeable blemish that might take some time to go away. Maybe.
https://www.quora.com/What-would-happen-if-we-fired-all-nuclear-missiles-on-earth-at-the-sun
Just in case anyone was wondering..
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