I received a greeting card, cake and a Mozart CD set from LH on my last birthday, delivered to my London address. All quite lovely actually!

And the birthday before that, I was actually onboard a LH flight (in F) and the Purser, FO and a couple of other crew members came by with a cake & candles after the main meal service. The cake was then cut and also offered to the other 3 F pax. I also received a card signed by the entire crew, which is something I've still kept!

Never got anything from BA.