I think the Colossus[1] predated the ENIAC but is still in line with your general theme of doing stuff for the military. In this case it was used for cipher breaking, not firing calculations.

You could argue that it doesn't really count though because it was only turing complete in theory: "A Colossus computer was thus not a fully Turing complete machine. However, University of San Francisco professor Benjamin Wells has shown that if all ten Colossus machines made were rearranged in a specific cluster, then the entire set of computers could have simulated a universal Turing machine, and thus be Turing complete."

[1] https://en.wikipedia.org/wiki/Colossus_computer