Lot of mathematicians have made several efforts to answer the question : are mathematics consistent ? Although we haven't yet had a proof of consistency, and surely we will never (Gödel second theorem), does this question has an impact on computer science ? I mean, does the work of those mathematicians, like Gödel's theorems or formal deduction systems, shows new answers or new problems in computer science ? What would happen in computer science if, for example, we found a paradox in set theory ?
Absolutely nothing will happen. Computers will still work, experts in machine learning will continue perfecting their tools, and even theoretical computer scientists will go on happily proving their theorems. Most mathematicians will also not feel much difference.
While we are trained to think that we are carrying out our proofs in ZFC, in practice proofs in mathematics and computer science are almost always carried out in English. Even if a contradiction is discovered in set theory, it is extremely likely that the axioms can be amended in such a way that the contradiction disappears and no proofs are affected apart from those in very specialized fields of a foundational nature.
More controversially, one can argue that most mathematicians argue in a nebulous system which actually does not rely on ZFC at all. Indeed, the average mathematician and nearly all computer scientists have little recourse to ZFC during their research life. Making this claim less nebulous, one can (and people do) argue that virtually all proofs in non-foundational fields can be carried out in weak subsystems of ZFC that are in less danger of being shown to be inconsistent. A nice reading in this general direction is McLarty's What does it take to prove Fermat's last theorem.
This would have a huge impact on computer science. All of CS is either a branch of mathematics, or built on the foundations of mathematics.
Every algorithm correctness proof or complexity bound is proven using mathematics. An inconsistency would shake the field to its core.
Moreover, through the Curry-Howard correspondence, proofs are isomorphic to well-typed programs, so a ton of type theory would go out the window.
That said, there is hope. There are multiple foundations of mathematics: ZF set theory, ZFC set theory, Martin-Lof Type Theory, Homotopy Type Theory, etc. Much of Computer Science has been done in all of these theories (particularly in PL theory), so if one is proven inconsistent, it may not mean that all is lost.
Let's assume that maths is inconsistent, and there is a proof that 1 = 0. Using that proof, you can prove that the runtime of any algorithm is zero.
That proof, however, doesn't change the runtime of the algorithm. So we would be in a slightly problematic situation: That we cannot use or trust proofs that are so complex that they can take advantage of this inconsistency. I think it is only very slightly problematic.
I think that the other Debaters answered the question concerning an impact of the consistency of mathematics on computer science.
But I have some comments to the following sentences:
Lot of mathematicians have made several efforts to answer the question : are mathematics consistent ? Although we haven't yet had a proof of consistency, and surely we will never (Gödel second theorem), does this question has an impact on computer science ?
Namely, I recommend the paper: T. J. Stepien, L. T. Stepien, "On the Consistency of the Arithmetic System", J. Math. Syst. Sci., vol. 7, No.2, 43-55 (2017); arXiv:1803.11072. There in this paper a full proof of the consistency of the Arithmetic System was published. This proof had been done within this Arithmetic System (of course, on the silent assumption that first-order logic is consistent).