While a number of battleships met their end at the hand of atomics at Bikini Atoll, likely the only dreadnoughts to carry nuclear weapons for tactical use were the Iowa class.
Those fast battleships “may have” toted such devices in two forms.
Between 1956 and 1962, the Navy had a limited stockpile of about 50 MK-23/W23 “Katie” nuclear shells for the Iowas‘ 16-inch guns, each with a yield of some 15-20 kilotons, with most ships of the class equipped to carry as many as 10 of these mushroom makers. Of note, Hiroshima’s Little Boy was a 15kt bomb.
USS Iowa, USS New Jersey, and USS Wisconsin had an alteration made to Turret II magazine to incorporate a secure storage area for these projectiles (the Nuclear projectile). USS Missouri was not so altered as she had been placed in reserve in 1955. This secure storage area could contain ten nuclear shells plus nine Mark 24 practice shells.
These nuclear projectiles were all withdrawn from service by October 1962 with none ever having been fired from a gun. One projectile was expended as part of Operation Plowshare (the peaceful use of nuclear explosive devices) and the rest were deactivated. USS Wisconsin did fire one of the practice shells during a test in 1957. It is not clear whether or not any of the battleships ever actually carried a nuclear device onboard, as the US Navy routinely refuses to confirm or deny which ships carry nuclear weapons.
Then in the 1980s came TLAM-Ns, the so-called nuclear Tomahawk cruise missile with its W80 150 kiloton warhead. First fielded in selected fleet units, only about 300 made were produced and the Obama administration dismantled them in 2010.
Below is a great video done by the curator of the USS New Jersey (BB-62) Museum, where he shows off the (possibly) TLAM-N related areas of the ship, including the panels, Marine guard post, and ABLs.