It depends on context, but "we're on the clock" means that we are being paid to do work right now. It is work time (rather than private time or rest time).

Hi Sinemn, You say "We are running late" (means: it's is late and you'll arrive late)

I would like to slightly disagree with the previous answer. As far as I understand it, "on the clock" means under time pressure, possibly with the idea of some third party watching whether you are late or not. It does not necessarily mean that you are already behind schedule or running late, but I guess if that does happen then it would be noticed, and possibly there would be consequences. My understanding of this phrase "on the clock" gives me an image of someone timing what I am doing with a stopwatch. It does not literally mean that, but I think the image sums up the idea.