Hi
For the last few months I've been trying to piece together who prints banknotes for the Bank of England and this is how much I've manage to compile.
Printers:
De La Rue established close relations with the Bank of England in 1724 and has been exclusively printing banknotes for the Bank of England since 2003.
Portals First received their order request to print notes for the Bank of England in the 1720s and it looks like they printed notes for them until Portals was bought by De La Rue in 1995 and I don't know what their operations were from then until their Portals had some independence under De La Rue to use the name again from 2018 to produce notes operate from De La Rue.
Waterlow and Sons, They did have some involvement in the printing of the 10 Shilling and pound notes from just after the war but mostly stuck to printing treasury notes.
Sources:
http://www.britishnotes.co.uk/news_and_info/features/
https://portalsinternational.com/welcome/history/