I did just that a few years ago at FCO and I wouldn't do it again, particularly with checked luggage.

I was on the last flight of the day to LHR, which was delayed by about two hours, so I figured I'd arrive at the airport accordingly. I'd OLCIed, but this was before mobile BPs, and I hadn't printed my BP before leaving for the airport as I didn't have a printer available. I got to BA's check-in desks at FCO at about the scheduled dep. time, but still two hours ahead of the actual dep. time because of the announced delay, and nobody was there. All the staff had packed up and gone for the day. Fortunately I only had a carry-on and I was able to print my BP at one of the machines. Bottom line, I wouldn't count on BA to pay for check-in staff's overtime at outstations in case of delayed flights, especially now that they're not even paying for lounge access or even fast track at several locations.