Concur with most that you should take the later BA connection. DL will initially go to LHR Terminal 3 (from what I've heard so far - though eventually all of SkyTeam is supposed to be in T4) and BA flights will leave from either Terminal 1 or 4 or 5 (depending on when you're going, BA flights are being moved around Terminals in stages) which means you will most likely have to take a rickety bus to the FCC (Flight Connection Center) if your BA flight leaves from T1 - a centralized connecting-passenger-processing facility (which will include UK security though 2 carry-ons are now allowed at LHR). After security there, you will be able to get your BA boarding passes (essentially check-in for your BA flight then - DL will not be able to do any check-in for BA except check your bags through).

If BA leaves from T4, there is a separate bus from T3 to T4 bypassing the FCC.
If BA leaves from T5, I have no clue how that will be done...