So, based on all the available evidence, it doesn't look like the "shock" of a nuclear blast itself, or any kind of significant shaking, will do anything of note to Yellowstone.
Well what about a powerful nuclear device like America's B83, a bomb that could release 5 quadrillion joules of energy in a heartbeat? Could that crack the crust just enough to cause a catastrophic collapse and subsequent depressurization event?
Nope. Even at its thinnest segment, the shallow magma chamber is around 5 kilometers (3.1 miles) or so beneath some fairly dense crust – and it takes a lot of energy to excavate it. As it so happens, people have tried, and no manmade crater that deep has ever been created.
To be fair, subterranean nuclear tests do have a history of generating fairly sizeable subsidence craters. During the Cold War, both the US’ “Project Plowshare” and the Soviet’s “Nuclear Explosions for the National Economy” both used buried nukes to see how good they were at digging holes in the ground. In total, 151 so-called Peaceful Nuclear Explosions (PNEs) were carried out by both sides.
The largest crater produced as part of Operation Plowshare was formed in Nevada on July 6, 1962. A shallowly buried nuke, equivalent to 104 kilotons of TNT, created a crater (named "Sedan") that was truly vast in terms of volume, but was “only” 100 meters (328 feet) deep.
The modern B83 nuke is about 10 times more energetic, TNT-wise, but that doesn’t mean a crater will be10 times deeper – that’s just not how craters work. A nuke at the surface is way above something called the optimal depth of burial, which is the depth required to produce the largest crater for explosions of a set energy.
Just look to the most powerful nuclear blast in American history, the Castle Bravo test in the Bikini Atoll on March 1, 1954. Coming in at 15 megatons – way more powerful than the B83 in use right now – this surface detonation, which took place in a coral reef, dug a hole in the reef only 76 meters (250 feet) deep.