You do NOT need a carseat on London buses, trains or black taxis. I don't think you need one in a mini cab by law, but we always ended up using one. Black cabs, I just sat DD on my lab or left her in the stroller and wheeled her in the back. Buses and Trains - either left her in the stroller or if it was crowded, folded stroller up and sat her on lap. A baby carrier will be even easier!
No carseat required on the plane either. They have special seat belt extenders that they give to lap babies for take-off, landing and turbulence. Otherwise you can keep her in the baby carrier or if you get a bulkhead seat, their are normally cots to put them in. They will not let her stay in the carrier for take off or landing, which sucks if she is asleep, so once you get settled in your seat, I would take her out of the carrier and get ready to put the belt on.