Originally Posted by Andriyko

You cannot combine two separate tickets. What you can do is change your SFO-LHR flight into SFO-LHR-ATH flight which will entail paying a fare difference, if any. If you present yourself at SFO with 2 separate tickets and you have luggage, the agent will only tag your bags to LHR and, yes, you will need to clear immigration control, pick up your bags and recheck them for the next flight. The best thing to do is to change your ticket.