$\overline{SAT} \in NTIME(subexp)$? Is it possible that $\overline{SAT} \in NTIME(\exp(n^{0.9}))$ ?
Are there interesting consequences of such containment?
Would it contradict the Exponential Time Hypothesis?
It is possible ;-)

It would give new circuit lower bounds.  Since you are making a pretty strong assumption this could follow from the seminal work by Impagliazzo, Kabanets, and Wigderson, I haven't checked.

If you use Williams' approach, tightened here, you get a lower bound of $n^{1+\Omega(1)}$ for a function on $n$ bits in the class E$^{NP}$.  (For this approach, a fast co-nondeterministic algorithm for satisfibility is sufficient, as you are postulating.)  Specifically, the latter paper shows that a lower bound for size $s$ follows from a satisfiability algorithm for circuits of size $O(s)$, which we can translate to a 3SAT instance with $O(s)$ variables by Cook-Levin.

It would not directly contradict ETH because that's for deterministic algorithms.