Was James Stuart the king of Scotland and the King of England at the same time?

It was at the point a United Kingdom.

His reign was a primitive version of a united kingdom, but James was a king of two realms, Scotland and England. He had been King of Scotland for 36 years before becoming King of England.

https://www.royal.uk/james-i-r-1603-1625

He tried to make a governmental union of the two realms, but it failed to take hold. Scotland proved easier for James to deal with, meeting his aims, but England had its own set of problems. Both countries also wished to remain their own sovereign state. The realms were separate, with their own parliaments, judiciary, and laws. James was James VI of Scotland and James I of England.

https://en.wikipedia.org/wiki/James_VI_and_I

James was ambitious to build on the personal union of the Crowns of Scotland and England to establish a single country under one monarch, one parliament, and one law, a plan which met opposition in both realms. "Hath He not made us all in one island," James told the English parliament, "compassed with one sea and of itself by nature indivisible?" In April 1604, however, the Commons refused his request to be titled "King of Great Britain" on legal grounds. In October 1604, he assumed the title "King of Great Britain" by proclamation rather than statute, though Sir Francis Bacon told him that he could not use the style in "any legal proceeding, instrument or assurance".

No it did not, from james onwards the king or queen of england was the monarch of scotland too, although it took until queen anne was monarch that it was made official.

Quite so, dear girl. You see, two Kingdoms are better than one, as they say in the movies. And three are better than two. Just sayin'.

I have supplied a link which will help you understand the situation.

Yes. They were two separate kingdoms and not united until a century later. (James I was also King of Ireland.) This is known as the Union of Crowns which began in 1603.

King Charles I was monarch of the three kingdoms of England, Scotland, and Ireland from 1625 until his execution in 1649.

From the Scottish perspective the present queen is the first to bear the name Elizabeth, but because there had been a Queen Elizabeth in England, she decided to be Queen Elizabeth II.

Stuart little wasn't the king of England! He was a little mouse with a large pet cat

The good news is that there are books written on the subjects of English history, Scottish history, and British history. Pick up any one of them, and you will encounter abundant references to James's having two crowns, one for each kingdom. Likewise, King Charles will be clearly shown as king of both England and Scotland.

As we recede into the new Dark Age, those books and, probably, all books will disappear into oblivion. But take heart. It may be only a thousand years or so (based on past experience), before mankind will recover the blessings of literacy, and the pen will once again become mightier than the sword.

Yes, he was King James I of England and King James VI of Scotland. The next two Kings were Charles I and Charles II of England and Scotland then James II of England/James VII of Scotland. When he was deposed his daughter Mary became Queen Mary II of both England and Scotland while her husband was William III of England but William I of Scotland. While Mary II was the second Queen Mary of both England and Scotland, the first Queen Mary of England was Mary Tudor while Mary Queen of Scots was the first Queen Mary of Scotland. Mary's sister Anne was the last Stuart monarch and the last monarch of the two kingdoms. She became Queen of Great Britain, uniting England and Scotland, in 1707. She was followed the the six Georges, William IV, Victoria and Elizabeth II. Elizabeth's title caused some concern in Scotland as there had never been a previous Queen Elizabeth of Scotland.