BA have been using battery powered Sennheiser ANR headsets for years on the retro Airbusses they acquired from BMI. I think they're all the HMEC 26, but in use they were frustrating (not sure if this has been fixed since it's been a long time since I was flying the Airbus for BA) because it was difficult to balance out VHF1, VHF2, and the intercom. Having said that, the same thing often happened with the headsets supplied with ship's power. Apparently they've changed manufacturers but no idea if this solved the problem.
As far as I know, the entire BA fleet uses ANR headsets.