For UA, ticketing is handled via a queue. Whoever makes the change to the itinerary is responsible for queueing the reticket, and then it's automated from that point forward, when successful. A failed reticket requires manual intervention. (It can fail if the operating carrier does not return confirmed inventory, if the payment method fails, etc.). I'd expect that all tickets in the same PNR would enter the ticketing queue together, but admittedly, I don't know whether or not there are any guardrails in place to ensure that happens. Tickets in the same PNR have to match in every particular (date, routing, fare class, flights, carriers) so there'd never be a situation where you'd need to reticket one but not the others. However, if the PNR got split, I could see that happening if the agent who made the change forgot to reticket other passengers. It's a really weird thing to forget, though.

You can only discover a problem by checking with the operating carrier and validating the e-ticket numbers, as described. Most airlines will show this online, but you can always call and ask them to read the ticket number to you, or you can read the correct ticket number to them and ask if they match. There won't be anything on the United side that says "we sent this information to the partner."

If something fails, the phrase I'd use is that "Airline XYZ doesn't seem to have the correct ticket number. Can you please push the ticket through to their system again?" I'd also try calling the operating carrier and see If they'll agree to let you read the new numbers (one per passenger) to them. YMMV on whether or not they're willing / able to enter ticket numbers manually into their PNR, though. You can also try a conference call between the airlines, although if both have long hold times, that can be a challenge.