[1] https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimi...
Apart from Voevodsky's work, there are several other big math formalization projects. Perhaps the biggest ones are the formalization of the Feit-Thompson theorem (done by Georges Gonthier and his collaborators, in Coq), and the proof of the Kepler conjecture (done by Thomas Hales and his collaborators, in Isabelle/HOL and HOL Light).
On the other hand, many of his collaborators are interested in the project precisely because it uses Coq and they like constructive mathematics. So the HoTT book only uses exluded middle when necessary (and carefully tracks where), and developing a computational interpretation of univalence is considered to be an important task.
Edit to add: So why would someone who doesn't care about constructivism want to use Coq? Because it's based around typed objects. Apparently, homotopy theorists had for a long time been aware that instead of basing mathematics on sets, and then constructing equivalence classes on top of those, you could take homotopy types as primitive.. something like basing mathematics on blobs rather than points. This was all considered theoretical and impractical--until someone noticed that the logic provided by Intensional Type Theory (as implemented in e.g. Coq) can be considered to do exactly this!
http://en.wikipedia.org/wiki/Homotopy_type_theory#Univalence...