FULL SERVICE airlines won't generally be petty enough to deny you check-in and boarding if you're a little late for a valid reason.
Ummm, maybe. Don't forget at many airports in the UK even if you could persuade the Check-in Staff to let you "have a go" with handbaggage only the airport authorities impose very strict time limits on the passenger presenting his/her boarding pass at one of the conformance desks.
At the end of a day everyone can come up with a "valid reason" why they should be an exemption to the flight closure time. As a result even the full service airlines have to have a cut off time, otherwise it would result in anarchy.