With UA (no status) I just called and said that their rebooking, that would get me in some 4-5 hours later than originally scheduled, didn't work for me and asked could they put me on the direct Delta flight instead. The agent (from their India call centre) put me on hold for a few minutes and in less than 10 minutes came back with the ticket reissued and a Delta PNR (in full F class). Oh and as the DL flight was 2-class in business vs. UA 3-class in F that I was booked in, I also got a refund of the miles difference back for business vs. first.

With DL (actually twice, once I was put on UA and once on SA) and with AA it was proactively so done by them (though with a bit of prodding for DL to act, as is always the case to get them to do anything).

In each case though there was an arrangement for the other non-alliance carriers to "accept __ [insert ticketing airline name here] tickets", as is normally set-up in case of a flight cancellation.