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There are Royal Mail boxes all over T5. The one airside is on the lower departure level nearish Louis Vuitton (follow the red sign for HM Customs). Landside there are two boxes each side of Zone E, so to the left of South Security and to the right of North Security Main Entrance. For obvious security reasons the aperture is only a few millimetres wide, OK for letters but not for packages.
T3 Post Office is no longer run by the Royal Mail, but it is open on Sunday mornings as well as the rest of the week. It can be used for packages, Amazon pickups and for banking facilities. There are some post offices in the vicinity of Heathrow which are open 7 day a week and extended hours, essentially they are inside Co-Op stores. Note that the Left Baggage stores at LHR, LGW and Paddington (the people who store luggage at the airport) have some relationship with courier companies such that you can send packages and parcels using DHL and other operators. Not cheap but certainly convenient.