Shtukas for $\mathrm{Spec}\,\mathbf{Z}$ This is a very soft and speculative question. Please feel free to downvote, close or delete it.
Studying the cohomology of moduli spaces of shtukas, Drinfeld proved the Langlands program for $\mathrm{GL}_2$ for global function fields, which was later extended by L. Lafforgue to $\mathrm{GL}_n$.
Peter Scholze has a conjectural section Shtukas for $\mathrm{Spec}\,\mathbf{Z}$ in his 2018 ICM report https://arxiv.org/abs/1712.03708 on arXiv.

Might this be used to attack the Langlands conjectures for $\mathbf{Q}$ (and how)?
What could this be used for?

 A: Scholze's "mixed characteristic shtukas" are already the basis for his attack on the local Langlands conjectures, so in some sense they have already found application to ``Langlands over $\mathbb{Q}$''. The basic strategy here is to transpose the ideas of Vincent Lafforgue's work on the global Langlands correspondence to shtukas over the Fargues-Fontaine curve. 
As for the global problem, it should be noted that Drinfeld's moduli spaces of shtukas are themselves function field analogues of objects that you have (sometimes) in the number field case, namely Shimura varieties. The application to Langlands comes through the fact that the cohomology of moduli spaces of shtukas "realize" the Langlands correspondence. The analogous hope for Shimura varieties is false; for instance we have Shimura varieties for $GL_2$ (i.e. modular curves) but their cohomology cannot capture all automorphic representations, e.g. those attached to Maass forms. 
The "universal cohomology theory over $\mathbb{Z}$" doesn't necessarily seem to solve the problem of being able to find all the expected automorphic/Galois representations in geometry, which is the fundamental premise for the work of Drinfeld/Lafforgue, but maybe Scholze has a grand plan for this. 
A: After their work on integral $p$-adic Hodge Theory, Bhatt, Morrow and Scholze realized a certain $q$-de Rham cohomology theory was underlying their $A_{\rm inf}$ cohomology in their paper. This seemed to have the nice feature of "interpolating" among all sorts of cohomology theories: $\ell$-adic étale, crystalline, singular.
He's just speculating about how to make this completely precise, and possibly show such cohomology theory, once constructed one day, is "universal". It is unlikely that this will have applications towards global Langlands over number fields, let alone conjectures on algebraic cycles. 
It simply seems to me Scholze just wants to understand "what's going on".
Merlin's comments and the answer by user84144 say it all
