While the RAW answer is to roll death saves in initiative order, the fact you're playing per-round initiative means you've already deviated from rules as written, and there is strong precedent to continue deviating from them here.
RAW, 5e combat plays like a board game. Once you've established player order through rolling initiative there's no real meaning to the "beginning" or "end" of a round, turns just move in a circle indefinitely. This is reflected in the design of spells and abilities. Nothing happens at the beginning or end of a round (even lair actions are pinned to an initiative number), everything is tied to either the beginning or ending of somebody's turn.
If you mess with this design, you also mess with this turn-to-turn timing. A simple example is the Monk's Stunning Strike ability, which if successful lasts until the Monk's next turn. With cyclical combat, this means the target will miss exactly one turn. With variable initiative it means they'll miss somewhere between zero and two turns (making the Monk want to roll their next initiative badly after stunning someone!).
Describing his own per-round initiative house rule, 5e designer Mike Mearls specifically mentions moving end-of-turn triggers to end-of-round to avoid these kinds of timing problems.
This leaves you with two broad options as DM:
- Continue to have the player roll initiative, and make death saves on what would be their turn. This adds more tension and danger to being unconscious, because once you reach two failed saves you are at the mercy of the dice as to whether your healer will have time to stabilise you in time for the third.
- Move death saves to the end of the round. This will give players more predictability, because they're already at the mercy of the death-save dice, and adding another one to the mix might be annoying.
I would go with the second one. Dying is already an unpleasant experience, and putting players in a situation where they suddenly want to roll badly, and that feat that gave them +5 initiative ends up killing them, just makes it feel even more unfair.