Sounds like you got Hillingdon Borough Council Social Careline. I'm not aware of this as a BA service, or how it could fit into their operations, which has a well developed programme for emergencies large or small. I wonder if the telephone number got reallocated to some other company? Medic alert bracelets are a big thing in the UK, there are several million of them being worn.
BA will post out Sunflower lanyards - that's a Newcastle number (+44191)