A federal judge prodded the State Department to quickly review a batch of 14,900 recently discovered emails as the controversy over Democratic presidential candidate Hillary Clinton’s correspondence while she served as America’s top diplomat continued to simmer.

Judge James Boasberg, in an order, set a deadline for the department to complete the email review by Sept. 22 to determine which ones contain sensitive government information and which are strictly personal conversations. That could pave the way for the emails to be...