Onward Bags on BA1
I can add a bit more info here and a slightly different view to that of cws (but given that I'm a checked baggage person and he's not, that's ok!)
On BA1 through checking starts at LCY who have no issues printing the correct baggage interline tags. On arrival at JFK I have a chat with Special Services (or whoever meets the plane, literally standing at the aircraft door) and ask them to confirm my bag will be interlined correctly. This results in a phone call / radio to the baggage unloaders (i.e. to a few feet downstairs) who then make sure my bag gets automatically placed in the "transfer area" for onward routing.
When following this method my bag doesn't appear on the baggage carousel (I have stood there to check!), but does appear correctly at my onward destination.
Just in case anything has changed very recently, I'll declare that I haven't taken BA1 for a little while as I've been fortunate to get F out of LHR.