The first person who comes to mind is Hilbert, although his work on quantum mechanics might not qualify as „very applied“. But Wiener and von Neumann certainly fit the bill. Many of the prominent pure mathematicians in the UK (and presumably elsewhere) were involved in war work and did some very applied stuff there—a prominent example would be Robert Rankin. Gröbner in Austria is an analogous case.
Added as an edit: the fact that many pure mathematicians have worked as code breakers is mentioned elsewhere but of course Bletchley Park is a rich source of examples. Turing‘s contribution there is common knowledge but he was one of several, including, notably, William Tutte and Peter Hilton.