Nukes at sea have been a reality for literally 5 decades, anon.
Carriers DO have a defense against attacks from nuke ballistic and/or cruise missiles. In fact, it's a 2-part defense. The first part is, simply, tiny-ship-big-ocean. Carriers routinely evade satellite detection by using weather fronts, known satellite tracks/capabilities, etc., for planning their courses.
Part 1 summary - if you can't find 'em, you can't nuke 'em.
The 2nd part of the defense is that the carrier is the fastest ship in the ocean, by A LOT. I've seen carriers "walk down" cruisers and destroyers at flank speed on calm waters and just blow past them. Their advertised top speed, like the cruisers and destroyers, is always "33 kts". It's actually quite a bit more. Cruisers/destroyers can reach speeds upwards of 45 kts, and conservative estimates of carrier speeds are roughly 20kts faster than that.
And since the "road" is pretty flat at sea, in every direction, the carrier has nearly unlimited options in which to skedaddle away from an impact point at a rate of speed that puts it over-the-horizon from the hyper-cruise missile, and well away from the impact point of a ballistic missile (BM's are aimed at fixed points - they don't have mid-course or terminal guidance to attack maneuvering targets).
Part 2 summary - carriers can get outa dodge fast.
Perfect defenses? No. Substantial? Yes.