Algorithmic randomness concepts and other concepts in algorithmic information theory (AIT) are defined primarily in terms of Turing machines. (If you look at the textbooks in this area by Li & Vitanyi, Calude, Nies, or Downey & Hirschfeldt, you will see that Turing machines are pervasive and central.)
Now, I am not aware that AIT has direct uses, in the sense that you can, say, use the theory of algorithmic randomness to build a random number generator, but the results in AIT show very clearly what you can't do--e.g. you can't define an algorithm that can generate an infinite sequence that is random in any ideal sense. AIT therefore constrains and clarifies what it is that practical algorithms can do. To give another example, it clarifies what you can know about the power of compression algorithms, which essentially exploit lack of randomness in the same way that AIT defines lack of randomness.
What I suspect is that in the past there would have been slower progress in understanding randomness in practical senses, and in file compression algorithms, without the clear conceptual framework provided by basic results and concepts in AIT--even though those results and concepts have no direct practical applications. That is speculation on my part, but there can be subtle and nonobvious but important ways that ideas influence the development of knowledge, and I suspect this is one area where that is true.
(For the record, I am at the level of an advanced beginning student of AIT who has also studied practical random number generation a bit, with a long history with various aspects of CS. There may very well be practical applications of AIT of which I am unaware. I have seen no evidence of that, though. (My primary area of expertise has more to do with the "development of knowledge" point I made in the preceding paragraph.))
Maybe it's also worth noting concerning Turing oracle machines--machines that are provided with a superior source of information to draw upon, an "oracle"--that a mathematician involved with Turing-machine related areas, Robert Soare, has suggested in a journal article that online databases are analogous to oracles. I still don't think that has practical applications, however, and I'm not sure the analogy is anything more than just an analogy.