Your TK ticket No. is def. not 14 digits. It is 235 followed by 10 digits. The info you got from UA (14 digits) is wrong; I'm surprised that the TK agent didn't realise that and offered to insert it in the TK reservation.

The issue is here that TK can't see the rebooked segments in the TK reservation (when rebooking done by UA). Hence, TK has no grounds for reissuing the ticket. Maybe you should call UA again explaining that TK can't see the rebooked flights in their system and consequently can't reissue the ticket - and ask if UA has a different way to relay to updated information?

I agree with the previous poster that you have a serious issue here because your (current) TK ticket doesn't match the new flights, i.e. the information from the UA agent that "all is good" is wrong. Ask them to dig into the ticket to see each segment (if UA is able to do that); then the UA agent should be able see the issue.