I remember that Bismarck used single-shot anti-aircraft guns of 3.7 cm, her only fully automatic weapons being 2 cm Flak. But what are the reasons Germany failed to develop something akin to 40 mm Bofors until very late in the war? Were these reasons purely technical or political as well?
The German army had a fully automatic 3.7cm Flak 36 in the mid-1940s, which formed the basis for a naval weapon during the war. The lack of urgency may reflect the common prewar underestimation of the danger of air attack to ships. Sixteen 3.7s probably seemed pretty impressive. War experience showed the need to improve AA armament in both quantity and quality.
The Bismarck was fitted out in a rush with a selection of what was available at the time. Even the heavy AA battery was a half and half mix of C31 and C37 mountings for the 8.8 and 10.5cm guns. The 3.7cm were a 1930 model with a rate of fire of about 30 to 40 rounds per minute. This weapon was a two-barrel, semi-automatic, single shot AA gun system. It was antiquated when it first went into production. By 1940 it was virtually worthless as an AA gun.
Bismarck had two different models of 10.5 cm AA guns, but I have never seen her described as having 8.8s. Is there a source for this?
According to navweaps "The original 10.5 cm Dop. L. C/31 mounting was the 8.8 cm Dop. L. C/31 mounting reworked to take the larger gun." These were what were on Bismarck (forward 4 mountings) and Tirpitz (after 2, later replaced by C/37). Apparently the designation Dop. L. C/31 was used for both 8.8 and 10.5 cm twin mountings ("doppel" = "double"). The 8.8 cm was used on the Deutschland class but later replaced by 10.5 cm. Using two different calibers within the same battery would be unusual to say the least; it would be interesting to see documentation thereof.
Seigfried Breyer's Battleships and Battlecruisers is a good reference work. According Koop & Schmolke's Batttleships of the Bismarck class the Bismarck was armed with 10.5 cm flak (no 8.8 and they were on the smaller ships), 3.7 flak and 20 cm flak. There's a specific book, Miroslaw Skwiot's German Naval Guns, that addresses the armanent in detail.