The last time someone posted about a UM, it was because DL sent the two kids to the wrong city (they switched their paperwork and thus switched their destination cities). So it sounds like all's well that ends well and your case was a piece of cake in comparison
My guess (and it is just that) is that DL has responsibility for the child while in their custody (I am sure something is mentioned in the paperwork the person dropping him off at the airport had to sign). They would probably put him in a hotel but I would think you would be responsible for the hotel cost (unless it was mx related).
One reason that the airlines don't allow UM's to travel on the last flight of the day is just for this reason - it gives the airline at least one more shot at getting him/her to the destination the same day.
As a former Red Coat...I've had to chaparone male (always same sex as UM) UMs when they've had to stay overnite in CVG. Arrangements are made with the hotel to have SEPARATE rooms with a connecting door. The UM is normally fed...and taken to their room. The UM is instructed that I would be in the adjoining room and to knock on the door ANYTIME if they would need something. In the morning the UM is escorted back to the airport, fed breakfast...and taken to the departure gate. Comair/Regional Elite would always pay for room and meals. Sometime we would have to overbook the flight the next day to accomodate the UM. This was in CVG....don't know how it's handled in ATL.