Depends on how seriously ill the pilot is and in which direction you are flying.
If he just can't get off the toilet or throwing up etc, and you are headed home, many crews would just continue. IF its a heart attack and seconds matter then a diversion would be in order to the nearest medical center. In either case, unless there was an extra crew member of the airline in question deadheading or a provided relief pilot, the remaining pilot would most likely continue himself. Most US airlines however would already have a 3rd pilot on board because flight and duty time limitations would require one the exception being NY or Boston to UK in the summer, then MAYBE there would be only 2 pilots.
I see it as extremely unlikely in any circumstances would a pilot-pax from the another airline be called to the flight deck.
Cheers
Wino