Is there a hidden link between the existence uncountable sets and the undecidability of the halting problem?
It's not a hidden link but one that has been made explicit using the language of category theory and also a very natural question to ask and study. There is a fair bit of material on the subject.


CS Theory question asking the same thing
Andrej Bauer's blog post about fixed point theorems and Cantor's theorem.
A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points, Noson Yanofsky, 2003. Gives you a gentle introduction to Lawvere's paper, below.
Diagonal arguments and cartesian closed categories, F. William Lawvere, 2006 republication of a 1969 article making these connections precise.
Incompleteness in a General Setting, John Bell, 2007
From Lawvere to Brandenburger-Keisler: interactive forms of diagonalization and self-reference, Samson Abramsky and Jonathan Zvesper, 2010. Extension of the Lawvere arguments to game-theoretic impossibility results.