Are there natural separations in the nondeterministic time hierarchy?
As far as I know, we don't know of such languages, or if we do, there is significant controversy about the "naturalness" of them. I know this isn't really a satisfying answer, but I can say:

(a) If you prove an $\Omega(n^k)$ time lower bound for k-ISOLATED SAT for every $k$, then you have actually proved $P \neq NP$. 

(b) One way you might hope to show that k-ISOLATED SAT is one of these natural problems in $NTIME[n^{k+1}] - NTIME[n^k]$ is to show that k-ISOLATED SAT problem is hard (in the usual, formal sense of having efficient reductions) for $NTIME[n^k]$. In fact this is the only way we know how to prove such results. But k-ISOLATED SAT is probably not hard in this sense, there are some very unlikely consequences.

The main reason is that k-ISOLATED SAT instances are solvable in $\Sigma_2 TIME[n]$, independently of $k$. You can existentially guess the isolated assignment, then universally verify (for all $O(\log (\sum_{i=1}^k {n \choose i}))$ ways to flip up to $k$ bits in the assignment) that none of the other "local" assignments work. 

Here is the proof of part (a). Let ISOLATED SAT be the version of the problem with $k$ given as part of the input (in unary, say). Suppose we prove that ISOLATED SAT requires $\Omega(n^k)$ time for all $k$. If $P=NP$, then $\Sigma_2 TIME[n]$ is in $TIME[n^c]$ for some fixed $c$ (the proof uses an efficient version of Cook's theorem: if there is a SAT algorithm running in time $n^d$, then any $c > d^2$ suffices). But we proved that there is a language in $\Sigma_2 TIME[n]$ that is not in $TIME[n^k]$ for every $k$. This is a contradiction, so $P \neq NP$.

Here is the proof of part (b). If every $L \in NTIME[n^k]$ could be efficiently reduced to a k-ISOLATED SAT formula (e.g., all $n$ bit instances of $L$ get reduced to $k$-ISOLATED SAT formulas of at most $f(k) n^{c}$ size) then $NP=\bigcup_{k} NTIME[n^k] \subseteq \Sigma_2 TIME[n^{c+1}]$. This would immediately imply $coNP \neq NP$, but moreover it just looks very unlikely that all of $NP$ can be simulated so efficiently within the polynomial hierarchy.