Flying from east coast airports, mostly on BA and DL, I have never had the green card taken at check-in. It will either be left in my passport or will be stapled to my ticket and boarding pass to be collected at the gate. If it was left in my passport, I make a point of handing it in to the agents.
Also, those airlines are pretty strict about checking passports at or very near the gate for UK bound flights. That may be because they are fined ~2000 UK pounds for any passeneger who arrives in the UK with incorrect documentation. However, the airline is supposed to positively tally boarded passenegrs with checked luggage because unaccompanied luggage is not allowed on international flights with very few exceptions.
[This message has been edited by james (edited 02-16-2000).]