On board not much (yet) is happening, as the focus of recycling (for LHR inbound flights) is cans and glass, but that is set to change in the not-too-distant-future; an initial attempt has been the reduction of the swizzle stick. What's taking a great deal more focus, and success, is the reduction of single-use plastics and coffee cups in the offices. A group of volunteers has achieved, in 2 months, a 50% reduction in cups' used at Waterside, and is also phasing out all single-use plastics from Waterside's canteens and cafés (not Waitrose though), with CRC and Engineering to follow. Getting rid of plastic on the planes, though, will be harder and not just because of lack of alternatives.
All this is my recollection & opinion and not BA's position.