The A-bomb won:

I wouldn't have thought it was necessary to detonate a 19 kiloton nuclear weapon to see what it would do to an airship, but that's just what the US Department of Energy did on 7 August 1957. Well, to be fair, the primary purpose was probably to test a prototype of the W30 nuclear warhead; the airship thing was just a bonus. The test, codenamed Stokes, was part of Operation Plumbbob, a series of 29 above-ground detonations carried out at the Nevada Test Site between May and October 1957. Statistically speaking, the radiation released into the atmosphere from Plumbbob would be expected to have caused 1900 civilian deaths from thyroid cancer -- a small price to pay for the knowledge gained, I think we'd all agree.

At least one other Plumbbob test involved US Navy airships: Franklin, or maybe Franklin Prime -- the sources I can find refer only to Franklin, on 2 June, but that fizzled and was repeated on 30 August. (The second and third photos here are from Franklin/Franklin Prime.) In 1960, the Bureau of Naval Weapons issued a report on the airship tests, entitled "Structural Response and Gas Dynamics of an Airship Exposed to a Nuclear Detonation". The abstract reveals that the aim was to see how an airship employed on anti-submarine duties -- the USN was still using these into the 1960s -- would fare after dropping a nuclear depth charge:

Four Model ZSG-3 airships, U. S. Navy Bureau of Aeronautics Nos. 40, 46, 77, and 92, participated during Operation Plumbbob to determine the response characteristics of the Model ZSG-3 airship when subjected to a nuclear detonation in order to establish criteria for safe escape distances for airship delivery of antisubmarine warfare special weapons. Restrained response data for 0.40-psi overpressure input were obtained during Shot Franklin with the ZSG-3 No. 77 moored tail to the blast. Unrestrained response data for 0.75-psi overpressure input were obtained during Shot Stokes with the ZSG-3 No. 40 free ballooned, tail to the blast, 300 feet aboveground. The first airship exposed to overpressure experienced a structural failure of the nose cone when it was rammed into the mooring mast, together with a tear of the forward ballonet which necessitated deflation of the envelope. The second airship broke in half and crashed following a circumferential failure of the envelope originating at the bottom of the envelope, forward of the car.

Sounds like airship-delivered nuclear depth charges were just not meant to be.¹

Image source: National Nuclear Security Administration/Nevada Site Office, here, here, and here.

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1. On the other hand, surely the atmospheric overpressure from an underwater detonation would be much less than an above-ground test would cause. On the other other hand, the airship in the top photo was 5 miles away from the blast; but if it had been nuking a submarine it would be much closer than that. I'm not convinced these tests proved anything one way or the other, but then I'm no nuclear weaponeer. [↩]