Cheapest way -

Fly INTO anywhere, including LHR, but must be on an AA flight, not BA. Taxes will be $2.50 assuming you book a direct flight.

Fly HOME from BRU or MAD on AA.

Consider low cost carriers to get in between as well as full service.
eg: easyJet, Ryanair

eg: Total tax per person for JFK-LHR/BRU-JFK is $54