This is actually a really intriguing question! Surprisingly, the etymology of the modern English "okay" is not as clear as one might think. There are several competing theories on the origins of the phrase, so I'm going to address those in turn.

Perhaps the most famous etymologist to study the word's history was Allen Walker Read, who published "The Evidence on "O.K."" in 1941. While Read notes that there are several alleged uses of the phrase in written documents prior to 1840, he argues that these were in fact other phrases, such as "ATT" and "O.R.," which were misread by scholars^(1).

Instead, Read advances the theory that the phrase became popular in the American English lexicon during the presidential election of 1840. He cites the formation of the "O.K. Club," which was an organization formed by Democrats in support of Martin Van Buren, also known as "Old Kinderhook^(2)." The phrase came to be used as a "rallying cry" by Democrats, especially the "radical" Locofocos, who used it as a watchword (similar to a secret handshake^(3)).

However, while Read's theory on the origins of "okay" is perhaps the most popularized, it is by no means the only one advanced by linguists. Frank A. Greco notes the existence of evidence discovered by Robert G. Gunderson "in which O.K. is used as 'all correct' . . . which antedated the O.K. Club by at least a month^(4)."

Greco and fellow linguist Mary Degges disagree with Read's interpretation, arguing instead that the modern English "okay" originates from the Scottish "och aye" and Irish "ough aye" which translate literally to "oh yes" and which were in use perhaps as early as 1600^(5).

While this explanation suggests the origins of the word "okay" are somewhat older than Read postulates, both Greco and Degges acknowledge that the word certainly entered popular use after Martin Van Buren's 1840 presidential campaign. However, they contend that the word itself was probably influenced by the mass immigration of Scots-Irish to the United States, who would have continued to use the phrase "och aye," eventually shifting to simply "okay."

*Other languages which include phrases similar to "okay" are the Mandinge "o ke" from Bantu and Wolof "waw-kay", which would have been used by some African slaves brought to the United States, and the Choctaw "okeh" which would have reached American colonists through contact with Choctaw natives. Unfortunately, I was not able to find compelling linguistic histories of these phrases, only tertiary sources (such as the Wikipedia article, which should not be considered nearly as reliable as academic sources). While one could speculate based on their similarity that they may have had some influence on the modern American English okay, Read's and Greco's theories are the most compelling on how "okay" entered our lexicon.

Today, "okay" is considered a "discourse marker" along with words like "oh" and "right" and phrases such as "you know" and "I mean^(6)." According to the Longman Grammar of Spoken and Written English, discourse markers are described as

inserts which tend to occur at the beginning of a turn or utterance and combine two roles, namely (a) to signal a transition in the evolving progress of conversation; [and] (b) to signal an interactive relationship between speaker, hearer, and message^(7).

Zarina Othman, who has studied the use of "okay" as a discourse marker extensively, adds to the above definition that

these markers are seen as 'inserts,' [which] suggests that they are items that should be syntactically independent, and also their absence should not affect the sentence meaning^(8).

So to put it simply, the phrase "okay" entered the American English lexicon during the electoral campaign of Martin Van Buren in 1840, but its etymological origins may be somewhat older than that. To answer your follow-on question, "okay" and "all right" can be (and are) used interchangeably as discourse markers.

If your hypothetical movie or series is set after 1840, then the use of "okay" is not anachronistic. If it's set before that, it probably is, but as I've mentioned above, there are competing theories as to when exactly the word came into existence in the modern sense - if your film is set in seventeenth century Scotland, for example, it might be reasonable.

*Edit 1.

Notes:

1. Read, Allen Walker. "The Evidence on "O.K."" The Saturday Review, 19 July 1941. Page 3.

2. Read, Allen Walker. "The Evidence on "O.K."" Page 3.

3. Read, Allen Walker. "The Evidence on "O.K."" Page 3.

4. Greco, Frank A., and Mary Degges. "The Etymology of OK Again." American Speech 50, no. 3/4 (1975): 333-35. Pages 333-334.

5. Greco, Frank A., and Mary Degges. "The Etymology of OK Again." Page 334.

6. Othman, Zarina. "The Use of Okay, Right and Yeah in Academic Lectures by Native Speaker Lecturers: Their 'anticipated' and 'real' Meanings." Discourse Studies 12, no. 5 (2010): 665-81. Page 666.

7. Othman, Zarina. "The Use of Okay, Right and Yeah." Pages 666-667.

8. Othman, Zarina. "The Use of Okay, Right and Yeah." Page 667.

Sources:

D. Biber, S. Johansson, G. Leech, S. Conrad and E. Finegan. The Longman Grammar of Spoken and Written English. Harlow: Pearson Education, 1999. ISBN 9780582237254.

Greco, Frank A., and Mary Degges. "The Etymology of OK Again." American Speech 50, no. 3/4 (1975): 333-35. www.jstor.org/stable/3088024.

Othman, Zarina. "The Use of Okay, Right and Yeah in Academic Lectures by Native Speaker Lecturers: Their 'anticipated' and 'real' Meanings." Discourse Studies 12, no. 5 (2010): 665-81. www.jstor.org/stable/24049862.

Read, Allen Walker. "The Evidence on "O.K."" The Saturday Review, 19 July 1941. Pages 3-4. http://www.unz.org/Pub/SaturdayRev-1941jul19-00003

Wow. Thank you so much! I'm Swedish my self but I'm very interested in languages and the history of them. I was actually watching the mini series "Saints & Strangers" on Netflix. It's about the Mayflower and the first settlers of New Plymouth. It was really bugging me when a woman repeatedly used the phrase "It's going to be okay". It just ruins the story.