Time complexity of languages that are polynomial time reducible to NP complete languages
You are right.  You can't draw that inference.  Given the assumption that SAT can be solved in $O(2^{\sqrt{n}})$ time, it does not follow that all NP-complete problems can be solved in $O(2^{\sqrt{n}})$ time.

For instance, the reduction from the NP-complete problem to SAT might transform a problem instance of size $n$ to a SAT instance of size $n^2$, so applying the SAT algorithm to that would take $O(2^n)$ time.  There are some reductions that preserve the size of the problem instance, and for those reductions, you will be able to solve them about as fast as SAT -- but as far as we know, not all NP-complete problems fall into that category.

You might enjoy reading about the exponential time hypothesis, which is the hypothesis that there is no such subexponential-time algorithm for SAT.  Folks have studied the consequences of the exponential time hypothesis in depth (as well as the consequences of the negation of the exponential time hypothesis).