Something like this would work:
LHR-DEL-HKG//PEK-HKG-BKK-HKG-(x)JFK-<S America>-JFK-WAS-JFK-MAD-TLV//ATH-LHR
Not sure the best way to do your S American destinations - you might need to buy two extra segments in addition to the 4 allowed to fit it in (which is OK so long as you stay under 20 air segments in total).
As number_6 says, you can't reenter the UK until the end, but you could fly to TLV via MAD, and then you'd need to make your way separately from TLV to Greece.