1. The Keisler-Shelah isomorphism theorem stating that two elementarily equivalent structures have isomorphic ultrapowers (proved using in $ZFC+GCH$ by Keisler in 1964, and in $ZFC$ by Shelah in 1971).
By the way, a left distributive algebra is a set with one binary operation $$* satisfying the left distributive law a(b*c)=(a*b)*(a*c); a*(b*c)=(a*b)*(a*c); the operation of conjugation in a group is an example. Here is an old paper of Laver about this topic; there is by now a large literature on the subject. 3. In some cases, the consistency proof of a statement S can be combined with some absoluteness argument to yield a proof of S (this was hinted at in the second example cited by Steprans). There are all sorts of absoluteness theorems in logic; the standard tool of the trade is the Shoenfield absoluteness theorem that shows that all sorts of consistency results can be translated into ZFC-proofs. 1 Here are my favorite examples of statements whose consistency was established and cheished before their proof. 1. The Keisler-Shelah isomorphism theorem stating that two elementarily equivalent structures have isomorphic ultrapowers (proved using in ZFC+GCH by Keisler in 1964, and in ZFC by Shelah in 1971). 2. Several algebraic results (normal forms, divison algorithms, etc) concerning the "left-distributive algebras of one generator" were first established by Richard Laver by assuming (very) large cardinals (known as (I3) in the literature). Later Patrick Dehornoy eliminated the large cardinal assumption in these results by giving a representation in the braid group. By the way, a left distributive algebra is a set with one binary operation$$ satisfying the left distributive law $a(b*c)=(a*b)*(a*c)$; the operation of conjugation in a group is an example. Here is an old paper of Laver about this topic; there is by now a large literature on the subject.
3. In some cases, the consistency proof of a statement $S$ can be combined with some absoluteness argument to yield a proof of $S$ (this was hinted at in the second example cited by Steprans). There are all sorts of absoluteness theorems in logic; the standard tool of the trade is the Shoenfield absoluteness theorem that shows that all sorts of consistency results can be translated into $ZFC$-proofs.