I've been reading Makers of Modern Strategy which goes into how mechanised warfare came about in Great Britain and France, both from a political/personal perspective and whereall they drew their doctrinal influences and designs from. It's a really interesting period I've never given much thought to and the political/economic/philosophical background of those countries' tank corps is a fascinating read.
Considering how much more advanced German armour at the onset of the second world war seemed to be, who and what guided the development of their tank corps and the tanks themselves? Was it purely owing to the restrictions of Verseilles or is it an extension of Prussian innovation? Were the Nazis more to thank for it or did it start pre-chancellorship?
It must be said, before I answer your question, its not that German Armour was technically superior or 'better' per se at the outbreak of WWII, but that its doctrine was more mature and well developed compared to its French and British counterparts. The Germans had a true combined arms theory of warfare, and while, yes, British and French armored formations often had Motor infantry and artillery attached, it was usually an afterthought and the training between such organic units was not to the same zenith as Germany's.
I'm going to try and answer your questions one at a time here:
Was it purely owing to the restrictions of Verseilles or is it an extension of Prussian innovation?
For interwar development, we go straight to the source. Guderian in his Achtung, Panzer! is keen to point out that experimentation with AFVs in the Reichswehr is minimal; and it isn't until the NDSAP take control that the budget, and more importantly, the government interest, is given to the Mechanization of the Wehrmacht. German tanks were lacking severely in terms of technical ability in 1933-1936; initial maneuvers and exercises actually had to be done with paper silhouettes of tanks, or something similar, tacked onto tractors and cars so that future Panzertruppen could learn formations, reaction-to-contacts, road marches, and other basic things. Versailles didn't just hinder the development of tanks, it outright banned it, and German development of their AFVs is a combination of indifference by the late Entente, innovation from far-sighted and clever Staff Officers, and a blithe willingness to gamble by the NDSAP.
Were the Nazis more to thank for it or did it start pre-chancellorship?
Nazis. As said above, they listened, engendered, and supported Reichswehr officers who wished to experiment in armored warfare. While other countries simply did not have the budget, or the stomach, to contemplate an expansion of their military and armor at this time, the NDSAP showed a willingness to fly in the face of the Versailles treaty and modernize and expand a neutered Reichswehr. The first production date of the PzKfw.I is in 1934; Hitler assumes full control of the German state in the year 1933.
The hyperlink leads to a thread where I discuss the cause of French defeat in 1940; and through answering that question I go into greater detail about the development of the German Panzer arm from the years 1933-1940. It also re-emphasizes how the Germans didn't have better armor pre- and early war, but more importantly better use of armor.