The Hood may have had better protection than the QE's or R's but when she met Bismarck it was not enough.
I have to say that I have never understood the RN policy on armour, they pioneered the original Dreadnought and the 15" ships so they must have known that other nations would follow building something similar in firepower. They also must have known the capabilities of the 15" & 16" shell, so why built a ship that is only slightly better protected that the R's and QE's? Obviously any hit from a large calibre shell is going to do some damage, but I wonder if Hood would have stood up to the tremendous battering that Warspite took at Jutland without sinking or blowing up. I realise that the later Washington Treaty made later battleship building difficult but again the RN built Rodney and Nelson with a pathetic speed that was even slower than the QE's, did the RN designers go to sleep between the wars or were they so complacent that they thought that no one would dare challenge them?
There are several issues at play here.
One is the typical capabilities of battleship caliber AP shells at the time these warships were designed. As Byron alluded to, pre-Greenboy British shells performed poorly. They often went off on contact if they went off at all. If they struck very far from the normal they often were rendered inert or they broke up. The relatively soft caps fitted over the heads of the shells often failed to do its intended functions. These were to stress the face hardened armour to enable the shell to penetrate the armour, and to protect the shell from being broken up by the face hardened armour. At that time a typical AP shell may not penetrate intact a face hardened belt of armour at any range.
The Greenboy shell changed the game. It could penetrate 14" of high quality face hardened armour intact at 16,000 yards, and it still worked given a more oblique impact. Nonetheless, as a practical matter the typical maximum a shell like a Greenboy could penetrate was about its own caliber.
Another factor at the time these ships were designed was the practical battle ranges of combat. At that time 20,000 yards was the maximum range that battle could be expected and the angle of fall of incoming rounds were less than 20*. This fact meant that horizontal protection schemes were made up of a series of protected decks intended to mainly contain splinters rather than "keep projectiles out", because shells would be fused by the upper belts and upper decks and explode before they reached the vitals.This was the function of the Hood's upper belts for example. Deck hits would be at such shallow angles that the shell would most likely scoop and the trajectory would carry it across the beam of the target until it exploded if it didn't scoop.
These facts meant that the vertical protection of a design could provide protection at just about all practical battle ranges if it was about as thick as the caliber of shell it was intended to defeat. 15" belt vs 15" shells was good. (or in the case of the Hood 12" sloped). A series of thin protected decks was good to 20,000 yards. By the late 30s 15" shells were far advanced over the Greenboy. The German 38cm had about 5" more belt penetration than that of the 15" Greenboy of 1920, for example (actually it had more belt penetration than North Carolina's new 16" gun). It still penetrated intact even when striking at angles up to 60* from the normal. It was not prone to scooping. It was difficult to remove the cap. Advances in weaponary had rendered the old generation of capital ship dangerously obsolete, even with modernization. It was lucky that warships like the Warspite, and the West Virginia, and the Nagato, were not put to the test by a modern opponent like the Hood was.
Given the Great Depression, navies could not really afford to replace old warships with new construction.
Another issue was that the Washington Treaty not only stopped new construction but also effectively stopped research and development of guns, armour, and shells . The Germans still conducted R&D secretly the entire time through Krupp, however.