Well, it might be interesting to have a look at the Bikini Atoll nuke tests! Here is a good summary: http://www.history.navy.mil/faqs/faq76-1.htm
There were two tests, one as an airburst that did very little (the raditaion was so negligible that the ships were bale to be reboarded within a day, and only five ships were sunk from this explosion (none of them battleships). The results of this test can be seen here: http://www.history.navy.mil/faqs/faq76-3.htm The next test was an underwater explosion in the middle of the assembled fleet, the center portion of which was largely made up of battleships and carriers. This test was much more destructive, with two battleships (Arkansas and Nagato) sunk, one carrier sunk (Saratoga), and several others. More importantly, the ships were all bathed in radioactive water spray from the explosion, which rendered them very radioactive indeed. However, the battleships, by virtue of their very thick armor, were largely protected from internal radioactivity, though of course their exteriors were quite hot!
After these tests, most of the ships were subsequently cleaned up and either towed away or sailed away under their own power, and were expended as targets later, as seen here: http://www.history.navy.mil/faqs/faq76-5.htm
In other words, nukes aren't particularly good weapons to throw at ships, especially not ships at sea (as against moored in a harbor), and battleships are the least vulnerable of all, due to the radioactive resistance of their very thick armor....
As for the supposed 'supremacy' of the carrier over the battleship, from every source I have read when examined closely, this is pretty much a myth! When you figure it took the combined efforts of 18 jeep carriers (the equivalent of SIX fleet carriers!), plus ground-based airpower to drive off a Japanese battlesquadron (at Leyte), and in fact, there was really not a lot of reason for that battlesquadron to turn away (other than the perception of its very ill Admiral, Kurita), and could WELL have pushed on and wiped out ALL of those carriers, this should give you some indication of what a proper battlesquadron can do, if it moves with determination! There were a couple other instances of this in the Solomons, where the Japanese Admiral (on board a carrier like Shokaku) would call off the battlesquadron, just as it was within striking distance of the American carriers (in fact, USS Hornet was 'finished off' by destroyers of this battlesquadron, and if they had just pushed on a few more miles, would have caught USS Enterprise and blown her out of the water, thus eliminating further US airsupport for Guadalcanal!). The reason for such withdrawals in all cases for the Japanese was the loss of their carriers, not their battleships or heavy cruisers (and this was the case at Midway, etc, etc, etc), which in fact were often in a position to have redressed the losses in no uncertain terms, but didn't because of the overcaution of their overall Admiral, rather than the Admiral on hand.... I really do recommend you get a chance to read 'A Battle History of the Imperial Japanese Navy' by Paul S. Dull (you can get a used copy from Amazon for just a couple bucks). Once you get a really good understanding of these various battles, you will get a better idea of how close some of these things actually were, with just a nudge required by a few battleships that could have turned a number of US victories into bloody and disasterous defeats...