I thought the UK started when James the first became monarch of both England and Scotland (the United Kingdoms)

Helen, it is MUCH more complicated than that. This link explains more than I knew before I checked.
Basically James VI (of Scotland) was invited to be King James 1 (of England). They were still separate countries under a single ruler. James called his realm the "Kingdom of Great Britain" and referred to South Britain and North Britain. He also introduced the Union Flag (it was called that at least by Charles I time) with the basic form as we know it. With the execution of Charles I, the partial union (of South and North Britain as well as Charles' head and body!) ceased, and a complete union was reformed by Queen Anne in 1707. I believe this was known as "The United Kingdom of Great Britain".
When the United Kingdom of Great Britain and Ireland was formed in 1801, the red diagonal lines were added to the Union Flag to represent Ireland (the cross of St.Patrick is actually from the arms of the Fitzgerald family).
Some people maintain that strictly speaking the "Union Jack" should only be called that when displayed on the jack staff on a boat, and it should be called the Union flag elsewhere. This is not true, it has been called either by the Admiralty, and Parliament officially adopted "the Union Jack" as the national flag in 1908.

I think the battle was just called Culloden (no moor).

Whenever I try to wade my way through the various layers of who was King or Queen to which bits of UK etc. when, I keep myself sane by reliving the Monty Python sketch from "The Search for the Holy Grail".
ARTHUR: How do you do, good lady. I am Arthur, King of the Britons.
Who's castle is that?

WOMAN: King of the who?

ARTHUR: The Britons.

WOMAN: Who are the Britons?

ARTHUR: Well, we all are. we're all Britons and I am your king.

WOMAN: I didn't know we had a king. I thought we were an autonomous

DENNIS: You're fooling yourself. We're living in a dictatorship.
..... A self-perpetuating autocracy in which the working classes--