As a general rule dictatorships tend to be extremely conservative. Science and invention of any kind require a freedom of thinking that is dangerous because it doesn't remain restricted to the desired topics.
Unless the could have found a military reason for colonizing space and it provided a tactical advantage.
BTW, to Roberts point upthread about a scenario in which the Nazi's win or the war is a stalemate one possible secnario is:
*The Normandy invasion fails
*Over the ~ 73,000 allied troops landing over 50% are killed
*The appalling number of Americans killed in the offensive sends Eisenhower back to the US in shame and public opinion swings against the War in Europe forcing Roosevelt to withdrawl ground troops from the conflict and focus on a completion of the war with Japan
*Congress with the ensuing public pressure passes a resolution forcing Roosevlet's hand - the US exits the European conflict
*The UK unable to, "go it alone," retreats.
*Hitler focuses all of his efforts on the Eastern front and facing extremely heavy loses Stalin halts the ground offensive at the Polish border.
*A stalemate between the USSR and Germany ensues. As a result, what we know today as France, Poland, Austria, and Germany becomes the German Republic for many decades to come.