Unfortunately, the Wikipedia list is incomplete for Scotland and both parts of Ireland. More complete lists are:
For Scotland:
Clydesdale Bank
Clydesdale and North of Scotland Bank
North of Scotland Bank
Bank of Scotland
British Linen Bank
Union Bank of Scotland
Royal Bank of Scotland
National Commercial Bank of Scotland
National Bank of Scotland
Commercial Bank of Scotland
For Northern Ireland:
Provincial Bank of Ireland
Allied Irish Bank
First Trust Bank
Belfast Banking Company
Northern Bank
Danske Bank
National Bank
Bank of Ireland
Ulster Bank
For the south of Ireland:
Munster and Leinster Bank
Royal Bank of Ireland
Provincial Bank of Ireland
Northern Bank
Hibernian Bank
National Bank
Bank of Ireland
Ulster Bank
Currency Commission Irish Free State
Currency Commission Ireland
Central Bank of Ireland
The list for England and Wales doesn't even begin to scratch the surface of all the provincial banks, two of which we already have notes from.
Former Numista referee for banknotes from Ireland, Northern Ireland, Scotland, Wales and Saint Helena.
Even after 1922, when the Irish Free State was created, there were banknote issuers that issued banknotes that circulated in both countries on the isle of Ireland. This ceased to occur in 1928, when the Irish Pound was created.
Chief Treasury of Wales & Black Sheep Company of Wales deserves to be listed under 'Wales' - which needs an independent listing.
The British Armed Forces notes need a separate listing.
We have all the banks required at present for Scotland, all parts of Ireland and Wales. We need a separate issuer for the British Military Authority notes (currently listed in the UK where they never circulated, see conclusions here) and we need the later NAAFI coupons moving to a new section in Exonumia. The referee for England is best placed to tell you what's needed there.
Former Numista referee for banknotes from Ireland, Northern Ireland, Scotland, Wales and Saint Helena.
British Military Authority & British Armed Forces notes should be listed separately under ‘Banknotes’, as they were issued by legitimate authority delegated by the British Government through the Ministry of Defence.