At least for Scotland the reason is historical, before the union they had their own pound, so they continued to mint their own coin.
Scottish notes technically aren't legal tender even in Scotland, but this because legal tender has a very narrow meaning in relation to court debts. They're technically promissory notes, the bank of Scotland has to hold reserves of sterling notes to cover them, although most is in special £100m value notes. Banks will accept them in England, some shops will not because they don't recognise them and think they're fake.