Base fuzes use the impact inertia shock to set them off, so they have to hit something hard enough or thick enough to initiate them; otherwise, the projectile acts like a solid shot until it hits the ground/water, which will set almost all such fuzes off (only a very few special anti-submarine fuzes were ever built that hitting water did not set off--they used either hitting the steel hull of the submarine (fat chance of that at most ranges!) or a depth-charge-like pressure-activated fuze (making them into rather tiny depth charges) and were not very effective, as you might imagine).
The fuze is armed on firing by the sharp acceleration force ("set-back") and, usually, also the spin of the projectile after it leaves the gun barrel. There is usually in most post-W.W.I designs a built-in mechanical delay to make sure that the projectile has cleared the barrel before the fuze is armed. Some fuzes used internal screw-threads that had to be spun a few turns to arm the projectile (much like an aircraft bomb fuze, but inside the fuze using a heavy weight that had inertia reduce its spin rate long enough to do the arming), others used ball bearings (Mark 23 fuze, see more below) or pivoting shutters that had to move outward using centrifugal force in a particular order to finally all get out of the way of the firing pin/detonator path, delaying arming long enough for safety, while others used mechanical locking rings and centrifically-thrown slides like a Chinese puzzle box that had to move in a particular order (U.S. W.W.II base fuzes, for example) to arm the fuze after set-back had released the locking pins preventing fuze arming prior to firing.
On impact, most designs had either a weighted firing pin that was thrown forward against the tiny high-explosive primer or, in non-delay designs, detonator capsule or vice-versa. A few designs, such as the early-W.W.II U.S. Navy Mark 23 Base Detonating Fuze (BDF) had a heavy spring that was cocked by this deceleration force and then threw the firing pin assembly back toward the detonator when deceleration ended (this is the fuze type that the W.W.II Japanese "diving" projectiles needed, but did not have!). The Mark 23 had some quality control problems and was not considered "safe" enough (the U.S. Navy seems to be the most paranoid military service in the world when fuzes are concerned, but they have had enough VERY bad things happen to very clearly justify this!!!!), so it was replaced by the ubiquitous Mark 21 BDF in AP and the larger base-fuzed "Special" Common projectiles by the middle of W.W.II. It usually takes about 0.003 second for the mechanical parts to function on impact, which gives the minimum possible delay from impact to final explosion with a non-electrical fuze.
The primer flash would set off the black powder delay element, which would burn for an instant (usually 0.006-0.03 second, depending on the delay desired, though the Japanese diving projectiles had much longer delays to allow a long underwater trajectory after the fuze was set off on water impact), then its flame on the far side would hit the more powerful detonator capsule, whose explosion would, in turn, set off the much larger booster charge, which then, finally(!!), set off the main filler charge. In non-delay fuzes, which had no delay element, the detonator and primer may be the same thing. The first large-scale delay-action fuzes introduced in 1911 by the German Krupp Company and Austro-Hungarian Skoda Company (these two companies seem to have had all sorts of mutual agreements and their W.W.I projectiles were very similar in many ways) had the combined primer/detonator and used a large black-powder delay element to set off the booster, made of picric acid (British Lyddite, which was so sensitive that black powder explosions could set it off), even though they had stopped using picric acid, replacing it with stacked pre-shaped blocks of TNT, a much less sensitive explosive, as their main filler charge. However, the detonator was still of a size to set off the booster, so it tended in many cases to literally blow the black powder delay out of its way and set off the booster with virtually no delay (or damage the delay element so much that it did not have enough power to set off the booster, resulting in a dud). These early fuzes tried to reduce the strength of the detonator blast by having a long triangular corkscrew tunnel between the detonator and delay, called by the British a "tortuous tunnel" when they examined such German fuzes after Jutland, but it did not work too well. The final concept of adding another high-power detonator stage between the delay and the booster and reducing the initial detonator to a tiny, low-power primer that could set off the delay without damaging it took some post-W.W.I genius to figure out. This made the fuze more complicated, of course.
Boosters were originally just large chunks of the same explosive used as the main filler, but packed tightly into a can (magazine or gaine) at the end of the fuze pressed tightly up against the detonator blast opening to ensure that the booster would be set off properly and, in turn, be powerful enough to ensure a good explosion of the main filler. As long as sensitive explosives like picric acid or black power or even guncotton (propellant powder used as a main filler charge in some late-19th Century projectiles) were used as main filler charges, this worked. However, when German/Austro-Hungarian TNT or U.S. Navy Explosive "D" or post-Jutland British Shellite (or sometimes TNT, which the British called "trotyl") were used, black power or picric acid no longer was powerful enough as a booster to set off these new insensitive explosives reliably unless very large boosters were used, which were reserved for large-filler, nose-fuzed projectiles only. Only very small boosters were allowed in AP and base-fuzed Common projectiles to strengthen the projectile fuze and body for armor penetration, so many duds or low-power explosions resulted with picric acid boosters (the most powerful "safe" explosive available through the end of W.W.I for booster charges).
Such poor performance was simply accepted by most nations to allow armor impact to not detonate the filler, as it did picric acid and, to a slightly lesser extent, black powder, which is why no delay-action fuzes had been developed earlier. The Japanese did not accept this and tried again and again during the 1920's to somehow cushion picric acid ("Shimose") fillers, but had to give it up as a failure in 1931, when they introduced their unique Type 91 Explosive, tri-nitro-anisol (TNA), which was barely able to be kept from exploding on armor impact when 33-40% of the filler cavity was filled with a wood, plaster, and aluminum "egg crate" cushion, but which was almost as powerful as picric acid (the insensitive others mentioned above were significantly weaker even when they exploded properly) and could still be detonated reliably by a small picric acid booster. Obviously, somebody powerful in the Japanese military had a fixation with post-impact effectiveness at all costs, even armor penetration, and nobody could tell him to "stuff it." :-)
The British and U.S. militaries both fixed their booster/main filler charge mismatch problem independently in 1928 by employing small amounts of the much more powerful, but previously dangerously sensitive, explosive "tetryl" in their boosters.
The U.S. used two tiny tetryl pellets in a pair of rocket-nozzle-shaped pits on either side of the end of their lipstick-can-shaped post-W.W.I base fuzes. These pits focussed the blast into needle-like jets (shaped charge!!) that could reliably set off even the U.S. Navy's ammonium picrate (Explosive "D") filler, which is the most insensitive explosive ever used in a gun projectile, to my knowledge.
The British flattened out several tetryl pellets into extremely thin flat wafers and stacked them, separated by thin metal foil layers, into a large booster can at the upper end of their base fuzes, each tetryl layer being small enough to remain safe (size is very important in explosives as to sensitivity!!), yet they would all be set off by the fuze under the bottom layer as the explosion moved up layer-by-layer through the booster. This caused a powerful explosion that could reliably set off TNT or Shellite (which was a very insensitive mixture of picric acid (70% usually) and a much less powerful and less sensitive explosive similar to picric acid in composition).
I do not think that the Krupp Company ever fixed its picric acid booster problem when using block TNT main filler charges, since the reliability of German Naval AP and base-fuzed Common projectiles during W.W.II, which always used such a filler and booster, to my knowledge, was rather poor as to explosive effectiveness, though obviously good enough in some cases, such as when HOOD blew up!
USA AA Time Fuze Operation
By Nathan Okun
AA Common--Mechanical Time Fuze (MTF) equipped (timed detonating nose-fuze) large-filler projectiles similar to HC, but usually without a base fuze and always using the MTF--were largely replaced during WWII by similar projectiles with VT fuzes, but time fuzes still were useful for shore bombardment and the older AA Common became a special-purpose HC projectile. These projectiles were developed for guns whose primary purpose was AA--the 5"/38 for example (HC projectiles of all sizes also had an MTF as an option, but also had a Point Detonating Fuze option and a Steel Nose Plug/Base Detonating Fuze option for other kinds targets--and most also had a VT fuze option when those were introduced in mid-1943).
The MTF time-setting device was a geared box-like mechanism controlled by the 5"/38's analog fire-control computer that had a hole in the front into which the projectile nose was pushed and its gear wheel would rotate the time setting ring on the projectile fuze to the calculated position (time interval after firing). Obviously, there was going to be a short delay between setting the fuze and firing the gun, set into the MTF calculations as a fixed offset, so a fixed rate of fire was a must if the pre-set projectiles were to explode anywhere near their targets (I assume that this delay was achieved by practicing again and again using a stop watch until the crew could load and fire at a very closely repeatable interval--firing too fast could be just as bad as firing too slow against an incoming aircraft, since the projectile would blow up at the wrong range).
VT fuzes eliminated the need for a fixed firing rate; the guns could
be fired as fast as possible as long as they were continuously being aimed
at the correct lead angle. This allowed a higher rate of fire and
more chance of hitting for each shot that was anywhere near the aircraft
target. Much more powerful AA capability resulted!!