When Barham was fitted with twin 4" AA mounts in 1938, three magazine spaces were fitted for them: one directly aft of 'B' 15in magazine on the centreline, and 1 to port and 1 to starboard outboard of 'Y' and 'X' 15in magazines right up against the torpedo bulkhead. This arrangement badly compromised the safety of the 15in magazines by placing large amounts of high explosives between them and the outer hull.
I think it is very likely that one or more of the torpedoes ruptured the torpedo bulkhead and vented into the port side 4in magazine space, causing a fire. The flames would of course rise vertically, but as the ship rolled onto her beam ends, the 15in magazines rotated until they were no longer horizontal to the 4in magazine, but were above. At that point the either the flames from the 4in magazine directly entered the 15in magazine, or some 15in cordite charges rolled into the burning 4in magazine; either scenario would result in the detonation of the after 15in magazines, but not until the ship was on her beam ends - exactly what is seen in the video.
Barham was likely torn to pieces by the explosion aft; her bows come out of the smoke inverted some distance ahead of the rising cloud, and stay on the surface for some time. As HMS Hood suffered a 4in magazine explosion which detonated her main after magazines, the video could explain why her bow is quite some distance away from the main centre section of the wreckage, and the conning tower close to a kilometer away. Hood's breakup was even more destructive than Barham's due to her high speed and the sudden loss of hull strength aft.
As always, the academics of analysis is tempered by the large loss of life and associated lives changed forever.