How can cargo be jettisoned from a C-124 in flight?
The C-124 had a opening in the rear, through which cargo (or paratroopers) could be dropped. From globalsecurity.org:


  The C-124 was also used for airdrops. ... In the back of the plane, the underneath of the airplane had elevator doors that opened up and the crew could drop cargo out, or paratroopers could jump out. 


Here is an image of the elevator doors in open position



The rear elevator opening of the Douglas C-124 Globemaster II where an elevator platform could be lowered or raised for bulk cargo operations (often the platform was removed so an interior hoist could directly move cargo) — photo by Joseph May; image from travelforaircraft.wordpress.com

Here is a C-124 using that door for an airdrop, presumably in Antarctica (I've put the image as I came across; I'm pretty sure that it's rotated)



Image from antarctica.recollect.co.nz

The crew must've used the rear door rather than the front one for jettisoning the nuclear weapon. The text of the official report reads:


  The C-124 aircraft was en route from Dover AFB, Delaware, when 
  a loss of power from the number one and number two engines was experienced. 
  Maximum power was applied to the remaining engines, but level flight could not be maintained. At this point, the decision was made to jettison cargo in the interest of safety of the crew and aircraft. Three weapons and one nuclear capsule were aboard the aircraft, and nuclear components had not been installed in the weapons. The first weapon was jettisoned at 4,500 feet altitude. The second weapon was jettisoned at approximately 2,500 feet altitude.


With the difficulties the crew was experiencing in keeping the aircraft level, it would only be logical that the rear doors were used, not the front.