In the first Austin Powers movie, Dr. Evil steals a 50 kiloton nuclear weapon, straps it to a drill and is going to fire it into the mantle to trigger volcanic eruptions world wide.
My instinct is that, compared to a lot of nuclear weapons in the megaton range, 50 kilotons is not really a lot, and using it to try to move rock (even molten rock) is not going to accomplish his goal. Would it even cause a measurable change in the pressure that far down?
Absolutely nothing.
For reference, here’s footage of a 5 Megaton nuclear device being detonated underground. No global eruptions, and hard for humans to even feel further from the epicenter - although, seismic monitoring by precision instruments is a known way of monitoring for secret underground nuclear tests.
For further reference, the 2011 Tohoku earthquake off the coast of Japan - the one that unleashed powerful tsunamis - released a staggering 3.9x10^22 Joules, or around 9.3 million megatons of TNT. While this did cause local devastation, it also failed to meet Project Vulcan’s goal.