I've recently gained the privilege of accessing a few review queues, but they never have any questions in them. Is this due to lack of moderators/the time that moderators have? Or is somebody sitting on the queue reviewing questions as soon as they pop up?
Review queues aren't necessarily involved with (diamond) moderators at all: the whole review process can pass by without needing diamond moderator involvement. The review queues are all about normal community members doing moderation tasks like monitoring first posts and suggested edits, voting to close or reopen questions, etc.
For the last few years, SFF has had a small group of people who are extremely keen on reviewing, to the point where almost everything gets reviewed by one of a very few users. The makeup of this group has changed over time, but you can pretty much see who the current people are by looking at the stats of the First Questions and First Answers queues, which are brand-new, having been first created on 26 August this year during the network-wide review queue updates. Between them, these two queues have had a little over 1000 items so far (in the last two and a half months), with over 400 reviewed by one person and nearly 300 by another one person.
So basically your second option: somebod[ies] are sitting on the queue reviewing questions as soon as they pop up.