I spent the weekend with my parents (both born in the late 1930s) and mentioned this thread. To my surprise they both completely agreed with the timeline suggested in this thread's initial post and would actually have judged the development to have been even faster.
Their reasons were utterly different from all of our speculations, though. They say that Wernher von Braun was the driving force behind all that. The Nazis themselves were not interested in rocket science at all. But von Braun wanted not continue his research from the theory to the practical use as a weapon and beyond that as a vehicle. Von Braun was famous and internationally acknowledged and so the Nazis did anything he wanted just to keep him (just for the prestige of having the best scientists).
When he saw no chance to continue in Germany (for reasons of there being no money, no raw materials and last but not least for having lost the war) he went with the winner - in a most literal way.
Without him, my parents reckon, the Nazis would have let the rocket programme die instantly as they had neither an interest in it not an understanding for the possibilities it offered.
The latter my parents deem to be a typical German character flaw. We have ingenuious ideas but never get through completely with them, so that others can exploit them.
They may be right with most German inventions like the fridge, the washing machine or the lipstick, but personally I am very glad that Oppenheimer and Co. built their nuclear bomb in America instead of over here, so that at least this weight is off my people's conscience.