Yes they would, if there is space available. You would be put on standby for it, and have a reasonable priority if it was due to an irrop (not sure if oneworld status counts for much? I suspect not). In the USA this is a very common way of life so it's quite slick. If you are Silver plus you can go upstairs to the Admirals Club airside in LGA where they are best placed to rebook you, or you could just hand your ticket over to the gate agent (less recommended). I guess the worst that could happen is that all services are full, but you'd still have connecting options from somewhere. I'd have a few hotel telephone numbers up my sleeve just in case, but otherwise best relax and go with the flow, there's always another aircraft.