So, all good advice, IMO, and @Steven Ayres summed it up best for me.
Accompanying jazz singers is one of the most difficult assignments I've experienced. Mostly because there seem to be two types of singers: 1) amateurs and 2) professionals. The pro's like to take a lot of liberty with the time but they know where it is and are counting on you to keep it while they walk all over it for the emotional affect they're seeking. The amateurs don't have the skills to count off the time accurately nor keep it once they start.
Seems a sad fact, but many singers don't play an instrument nor do they spend any time with a metronome, so, really, it's unreasonable to expect better.
What can you do? As Steve said, count it off for them. Providing them with a 4 to 8 bar intro helps a lot, but in many cases, you'll need to "nod" them in at the end of the intro.
Lastly, jazz tends to be about a static, consistent groove from start to finish, but singer's tend to be more Broadway, where the groove isn't the be-all-and-end-all and instead, the time shifts for emotional affect. Sensing that and playing to it without a rehearsal is a really tall order, IMO. I do know a handful of brilliant bassists who manage to pull it off, but I can't do it and don't know how they can.
Another thing that can help, and I encouraged when I ran a jam is, ask the singers for a recording of them singing the tune, or, if they don't have that, a reference recording that they're working from. When you hear that Disney pause, you'll know it's coming, even if you don't know exactly how long.
Last, it's jazz; a live performance, unrehearsed; you'll have to decide of the value if having the singer do a muddled tune is more valuable than having a crisp groove for the gig.