I'm not sure all of this discussion of interlining or separate tickets really matters that much or would have actually resulted in a different outcome for the OP.

Even if it was one ticket, the OP would still have had to collect bags and recheck - let's assume they would have gotten to the counter at 7:20 given they said they got their bags at 7:15 and would have needed to walk through customs, which sometimes has a line at ORD.

The flight departed from B11 at 7:45pm - so even if they dropped their bags, they would have needed to run up to the train, wait for the train, take the train, go through security in Terminal 1, and then walk to B11 in 10-15 minutes. I mean, maybe if you hit it just right, walk fast, know ORD well, and have no lines in security but I think 95% of the time, for most people, no way they are making that.

It sounds like UA rebooked them for the next day with no penalty, which is exactly what would have happened even if it was one ticket. And the interlining bag thing really doesn't come in to play with any re-accommodation

I don't check bags but I've heard ORD international bags can take a while. It is also not the easiest from a connecting perspective with the train. But to answer the OP's original question, probably 3 hours is safe - they would have been fine with that in this case