Well..... I shall answer myself!
Having got intrigued by this, I have just phoned The Environment Agency to get the latest info!
For a visitors licence, you do not need to give a BSS number. You just need to declare that your boat conforms to the Boat Safety Scheme.... which of course mine does, to the best of my knowledge!
Reading still does NOT sell visitor licences, but SOME locks in addition to Teddington are! (At the moment, these are Boveney, Shiplake, Sonning, Caversham, Mapledurham and Culham). More are to do so in the future. However, you cannot just buy a licence on foot, you need to be with the boat so they can see it, (whether on the trailer will do, I didn't ask)!
Second however…. at the moment if you launch away from one of the prescribed locks, you will receive a 'no licence' notice from the first lock you come to. When you subsequently buy a licence from a prescribed lock later on your travels, you will need to ring Reading and "tie the two up". That way, you won't get the stroppy letter at the end of the year!
All clear now then?
Right, when do we organise a 4 knot rib raid to Henley then to scare the s***t out of the locals!