Not clear that the order in which things are listed within the Start of Your Turn section reflects the order in which they occur in that phase. Ongoing damage is listed in there first, saying it happens now, then regeneration is listed, saying it happens now, but there is no explicit indication that one "now" comes before another "now."
I don't think it's a problem as long as the ruling is consistent, whether that means consistently following one order or consistently letting the individual decide. Ongoing first can get you killed, but at the same time if you are about to regen more than the damage you have sustained, taking the ongoing first is beneficial so that you can regen it. I would prefer a fixed order for everyone to a choice because, as a DM, I would like not to have think about which is better for the monsters. Then again, regen isn't so common that it matters that often.