I really can't see it. Aircraft carriers are simply impossible to hide and too vulnerable to missiles.
The Russians (et al) can see where are carriers are at pretty much any given time. So, in a Russian first-strike scenario, they just have to hit the general area of the carrier with a few nukes and they'd be pretty sure of a kill. Even if they didn't kill the carrier, the nuke effects would make air operations effectively impossible for a few dozen critical minutes.
I think the only time they could have been a significant player in that role was before lots of IRBMs/ICBMs, but only if the aircraft were long enough range that the carrier could stand off a LONG way. Not only did the development of missiles nix that possibility, but long range bombers made it somewhat irrelevant to launch from a carrier.
I guess they COULD contribute, though, as a quick strike capability at the start of a war (before they die).
IOTL, the US Navy developed the Douglas A-3 Skywarrior and the North American A-5 Vigilante as carrier-based aircraft capable of carrying strategic nuclear weapons. However, these aircraft had the misfortune to enter service at the same time as the first ballistic missile submarines, and thus were never used in their intended roles.
Your challenge is to have carrier-based aircraft the dominant naval component of the nuclear triad.