I know the Nazi's had business relations with IBM's subsidiary Dehomag, but I was wondering more generally if the Nazis were aware of Turning's theoretical mathematics and the beginnings of computer science.
Yes, there were German business enterprises that produced early electromechanical computers in the WW-II era. In particular, the mathematician and inventor Konrad Zuse created a series of digital, electrical computing devices beginning the 1930's, including the ability to store a program, and a programming language (Plankalkül) which foreshadowed many of the features of modern programming languages.
During the war, his work was sponsored by the German armed forces, where his machines were used for solving aircraft design mathematical issues, so to that extent, you could say the Nazi's were aware of the concept of electronic calculating devices contemporary with that of other nations, but at least for Zuse, it was developed independently.
After the war, Zuse founded Zuse Computers, where he progressively developed electronic computers for business and scientific systems. The firm was purchased by Siemens in the 1960's.
Addressing the issue of Zuse's knowledge of Turing's work, no Zuse developed his ideas and computers independently of Turing, nor was he aware of Babbage's work.
Sources:
http://www-history.mcs.st-and.ac.uk/history/Biographies/Zuse.html
The Nazi government provided some support to Konrad Zuse. They also excused him from military service so that he could continue to develop his computers. He started working on his first computer, the Z1, using his own funds in 1934. Later he received funding from the Aerodynamic Research Institute for his Z2, Z3 and Z4 (though the Z4 wasn't completed until after the war). His computers were used for some computations for aircraft design, but were not a serious contribution to the German war effort. He continued his computer work after the war.
As for Turing's work in the 1930's, and that of other computer science pioneers like Alonzo Church (Turing's adviser at Princeton), their papers were openly published in the 1930's. Apparently Konrad Zuse was not familiar with them though.