Why would $NP^ {SAT} \subseteq P^{SAT[O(\text{log }n)]}$ imply that $PH \subseteq P^{SAT[O(\text{log }n)]} $
Simply put, even a weaker claim of $\mathrm{NP}^\mathrm{SAT}=\mathrm{NP}^\mathrm{NP}\subseteq \mathrm{P}^\mathrm{SAT}=\mathrm{P}^\mathrm{NP}$ would imply PH collapse to the second level (so even the consequence is a little bit weaker, you can adapt to your case).

We have $\mathrm{P}^\mathrm{NP}\subseteq \mathrm{co}$-$\mathrm{NP}^\mathrm{NP}$. So, from the assumption, we deduce that $\mathrm{NP}^\mathrm{NP}\subseteq \mathrm{co}$-$\mathrm{NP}^\mathrm{NP}$. As a result, PH collapse to the second level.