Yes, both tablets have multiple saved Wifi access points, but here at home only the home Wifi and the PAWs.
Anyway, this power reduction thing had occurred to me, and it appears there may be a power mismatch somewhere between transmit and receive. I've been experimenting.
It is possible to get the Nexus to connect but it has to be at the correct distance from the Pi. On battery, this is no closer than about a metre and a half from the Pi, and no further than maybe 3 metres. On mains (Pi RS PSU), this is no closer than about 2 metres and no further than half way down the stairs, whatever that is including attenuations due to house bits, I don't have enough clear space in the house to test the free air range!
The software I used that worked I marked as being 2016-03-07 which was before the power reduction? I'm writing a new card with that software now so I can put in the licence to experiment with it.