I have been in that situation four times that I recall, each with outsourced crew, but I don't know whether they were applying their own policy or UA's (odds are it was UA's because the process was the same at separate airports).

Check-in would open at the regular time, most check-in positions would also close at the regular time, with one or two left open for passengers showing up later on account of the delay.

FYI, each occurrence I was involved in was in Europe.