The BoE ones are the "real" government-issued ones. The interesting ones are the genuinely private ones by the Scottish and Northern Irish banks. Including my favourite ever, the Northern Bank portrait-format space shuttle: http://www.polymernotes.com/northireland.html

The current legal basis is the Banknotes (Scotland) Act 1845: https://www.scotbanks.org.uk/history/banknote-history.html