Bascially BBJ is the line reference (Balham to Beckenham Junction Line?)
1265 is a reference number for that bridge
5m 02ch is the distance in miles and chains from somewhere, maybe from one end of the line or from London?
For Openstreetmap, I think you should tag the bridge with the reference number, using http://wiki.openstreetmap.org/wiki/Key:ref
Maybe the line reference could be added as something like ref_line=BBJ?
I’m not sure if the distance is worth adding to OSM, it should be obvious from the map.
I’m tempted to say that the unique reference for the bridge is BBJ1265, rather than use two separate tags.
Using ref= is OK for railway bridges, but what about references for bridges that have A or B roads crossing them? Won’t the ref tag already be in use? Perhaps bridge_ref= in those cases?
I just looked back at some photos I took when on a walk back at the end of January where the route took me under two bridges beneath a railway line. The older sign didn’t include the line reference (though the web links helped me guess what it should be), then the second photo did include the line reference which matched what the website had suggested for the first one. (MAH1050 and MAH1057). I can’t remember whether I tagged the bridges at the time, but will perhaps go back and add the tags now.