I think the answer is no. BA have telephone/online checkin if your are flying in one of the premium cabins (or are a member of the BA Executive Club), but I believe they are about to discontinue the telephone checkin side of things (great shame, as this was excellent). The online checkin is notoriously poor at checking in parties of more than one traveller - it's fine if you are flying by yourself, perhaps on business, but next to hopeless if there are a group of you. This may change when BA change their Executive Club arrangements on July 1st, but I doubt it.
Regards
Rob