I am putting a pair of old road tires on my commuter bike.

I noticed one tire has started to wear flat, while the other is still rounded and shows very little wear.

I am guessing that the worn tire is from the rear, but am not sure?

Either way should I mount the worn tire on the front or back?

The worn tyre is from the rear, I would put in on the front now so you can change both tires at the same time when it is completely worn. If it is worn so badly that it gives you trouble with flats it might be better to bin it and get a new one.

I would strongly recommend NOT putting a worn tire on the front. Traction on the front is much more important than traction in the rear, and a front flat is much more dangerous than a flat in the rear. The best course of action would be a new tire for the front and the old one in back, but if you don't want to buy a new tire, but the older one in the back.

Leave the fresh tire on front., since a failure there is more likely to lead to a crash than one in back.

When the rear tire shows cords, breaks cords and gets lumps, gets side-wall damage, or starts to flat unacceptably often put a new tire on front and move the previous front tire to the back.

As others have said, the center flat tire is probably form the rear, and that's where it should stay. Under normal conditions, I would never remove a tire and switch it to the other wheel. Front and rear tires wear differently, and in a way suited for where they are. why mess with that?

The main and immediate concern should be handling. A tire with a flat section doesn't track as well as a tire with a well crowned profile, so putting the old rear up front might lessen steering response and contribute to high speed wobble.

Even though the OPs decision doesn't involve removing a working tire since they're both off already, I'm suggesting that he remount them where they were and where he would otherwise have left them.