If you do not have a compressed air supply:
The suggested drawback of divers hemorrhaging out through mouth and nose because of pressure differential is admittedly colorful and dramatic, but absolutely will not happen if you're not giving the divers compressed air. Reason: diaphragm and intracostal muscles are not strong enough to inflate the lungs against a pressure differential, so your divers will not descend while breathing unless they have compressed air. I know this from personal experience: in my teenaged ignorance I tried to see how deep I could go with an adequately rigid tube. It turned out to be a pretty trivial depth, a few feet IIRC.
If you do provide compressed air:
It's still going to be highly questionable.
The likeliest injury would be air embolism, which is often fatal (and frequently causes brain damage when the victim survives.) This happens when ascending with compressed air in the lungs and (naturally enough, if you haven't been trained otherwise) holding your breath all the way to the surface.
As the air in your lungs expands, it tends to rupture the alveoli and force its way into the bloodstream. Neurological symptoms (up to and including death) are caused by bubbles blocking blood flow to the brain.
The biggest gotcha about air embolism: the maximum expansion happens closest to the surface. In other words, the danger increases as you almost get to to breathable air. :-(
Since the only treatment for this is a hyperbaric chamber, your divers in Roman times would be unlikely to receive treatment. Yes, hyperbaric chambers might well be invented - eventually - to treat the commoner malady of the bends, but it's kind of hard to see the engineering thought progressing like that. Romans were good engineers, no question; but they were not much for theoretical physics. And, since this issue is explained by Boyle's Law, which didn't emerge in our world until the 17th Century CE, you would really need to hack your timeline of scientific discovery...