I suggest that the OP keeps monitoring if award seats become available closer to the departure date (literally from a few days before) and in the affirmative, the OP can contact UA for rebooking.

Relying on TK to rebook the pax on the day of departure or enroute is not going to fly. As soon as agents realise that this is an award ticket issued by others than TK the OP will be referred to contact the issuer of the ticket for rebooking. TK will refuse to touch such tickets (unless in IRROPS situations).