whyamidoingthis
The United Kingdom of Great Britain and Northern Ireland would still be the only united kingdom in Great Britain and Ireland, and so it would be the definitive article. Taiwan is still officially "The Republic of China," just as South Korea is "The Republic of Korea."
The Kingdoms of England (inc Wales) and Scotland combined in the Act of 1707. After that, there was no England and no Scotland, it says so in the Act. That United Kingdom then entered a union with the Kingdom of Ireland to form the United Kingdom of GB & I. Scotland was not a party to that act.
In 1921, there was such a thing as (lower case) southern Ireland, a part of the UK which gained its independence as the Irish Free State, which became the Republic of Ireland in 1949, not to be confused with the Irish Republice which was the unrecognsed 32 county republic unikaterally declared by Snn Fein in 1918 and which SF continued to claim to be the sole legitimate govt of until 1999.
(See also Northern Ireland vs the north of Ireland.)
As a seceding state, Scotland would by default start with zero liabilities and zero assets save the land and territorial waters of Scotland. Just what would constitute "Scotland" would be a matter for the UK, which could (and in all probability would) keep Rockall and RNB Clyde, and also possibly other territory.