Having flown both DL and BA into FCO, I can confirm that House's info is correct - you will both be arriving into the terminal C satellite.
It is possible to get out of the immigration herd and back to either the satellite or the main part of terminal C. One simply stays on the SkyBridge train after everyone else has gotten off. If there isn't a guard to shoo you off, you stay on board while one set of doors close and the opposite side opens, allowing the outbound pax on. This puts you in the normal "departures" stream. Alternatively, you can walk out of the newly opened doors and proceed to the main departures area in terminal C, where the BA lounge is located.
If you stay in the satelite, you can try to talk your way into the lounge. DL used to use the L'Anfore lounge in the satellite - not sure what they use anymore. As a PM I've also used the AZ lounge adjacent. Neither is anything special. The BA lounge is much better, but may not be used by arriving pax.
Suggest you arrange to meet in the lobby bar at the Airport Hilton, which is very pleasant and not too far of a walk. If cost is not an object, you can also arrange for a day-use room at the Hilton and have a shower and change of clothes if desired. I've actually used the FCO Hilton as a base of operations when doing business in the outskirts of Rome (anywhere outside the GRA)...there's actually less travel time involved when one doesn't have to fight the central Rome traffic. It's a very nice hotel.
[This message has been edited by DFW DL (edited Dec 07, 2003).]