@Ahkai
It's a bit of a complicated process. In a very simplified manner, when the idea of banknotes took hold in the UK [Scotland from 1696], there were essentially no restrictions on who could issue them. The Bank of England itself was, at the time, only allowed to operate in London which meant getting official BoE notes outside of London was tricky, so the local banks simply issued their own. There was no central bank of Scotland either - the Bank of Scotland was probably the closest it came to having one but it never became the central bank (side-note, it still exists a commercial bank).
However, one of the problems which was suddenly an issue was when a local banknote was used outside of it's normal local area, the bank (or business) which eventually accepted it would then have to travel to the local bank to exchange it for actual coins - which at the time were largely in precious metals - or at least genuine BoE notes. But in that time between the local note being issued and someone going to collect, some banks may fail or simply not have the assets to back up the note (as restrictions weren't too tight), which made accepting them risky PLUS you had to pay someone to go and collect the coins, then run the risk of that person being robbed or even running away with your money. So then the other banks started only accepting non-local notes at a lower than face value - I have recollections of reading somewhere up to half the value in extreme cases.
Naturally, this upset people because no-one likes being told your money is worth less just because you've moved about in the country. So the UK Government stepped in (most notably in 1844, though other pieces of important legislation was passed in the 1700s & early 1800s) and required that all issuing banks had to be able to back up their notes with precious metal coins, and also made it so no new banks could issue them (and a bunch of other legal stuff around mergers/closures etc). They also allowed the BoE to open branches across the country, making official notes much more accessible. This helped ensure that most local banknotes could be trusted & traded at equal value to a BoE note. The issuing bank HAD to have initially bullion coins somewhere (and later expanded to also allow BoE notes), so it was pretty safe.
Over time through closures & mergers, Wales & England had no commercial banks left who could issue local notes (1908 for Wales, 1921 for England), though a handful continue to exist in Scotland & NI.
As banknotes became a more accepted part of everyday life (in part helped to the centuries of declining value of the pound), the amount in circulation just kept growing. Now the BoE only issues banknotes to the value of £50, nothing higher. Imagine you're an issuing commercial bank and you've got 100 million pounds of your notes in circulation. You would have to store 2 million £50 notes which is just a huge issue. Security, logistics, even storage space. Security is an especially big risk. If someone stole all those £50s, they could find their way into circulation or the black market. If someone steals that 100 million pound note, they can't exactly just cash it in or use it in trade. There's probably so few “active” notes in existence, they probably have a ledger which tells you the owner of each note.
If there was ever a run of one of these commercial banks, what would happen is the commercial bank would take it's 100 million note to the BoE. The BoE would then break it down into smaller official BoE notes for the bank, which the bank could then offer to it's customers at 1:1 parity.
I believe that it is the law that they have to have enough BoE notes (or coins or other tangible assets) to cover all of their notes. This protects the customers from any run, and creates stability in these commercial notes.
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P.s. I use the UK a few times here. In the earlier parts of my reply, I mean The Kingdom of Great Britain or even Scotland & England as separate nations, but I use UK here for simplicity and to avoid further confusion around the exact political situation. The UK didn't exist until 1801.
Also, this less-than-face-value issue wasn't just a UK thing - it was an issue for many nations. Most simply declared that commercial banks could not issue notes and phased them out, or a new nation replaced it with a new currency which could only be issued by the central bank.
As this was a summarised reply (hard to believe looking at it, I know 😛), I have missed out a lot of detail & technicalities. But that should give you a reasonable idea as to why such notes exist, and how they're used. If I've made any mistakes, I do apologise, been a while since I read up on this technical stuff.
