Actually Yes, James likely did have a thick Scottish accent. The most infamous work on James I was Anthony Weldon's Court and Character of King James first published in 1650 (Some scholars have expressed doubt that Weldon actually wrote this work. Whether Weldon actually wrote it is beside the point for this argument, as it was intended to be a genuine criticism of King James and his court). In the work that is filled with hyperbole, rumor and outright slander, Weldon mentions the King's speech. James's "tongue too large for his mouth, which ever made him speake full in the mouth". While older historians took this as either truth or as slander intended to make the king look like a babbling idiot, recent research has suggested an alternative. Instead of having a tongue too large for his mouth, it is more likely that James simply spoke with a heavy accent that the English though uncouth, slovenly and perhaps even that his tongue didn't fit in his mouth.

Francis Bacon, writing sometime in 1603 stated that the King's "speech is swift and cursory, and in the full dialect of his country." Bacon was among the first to meet with James during his progress to London for his coronation. Jenny Wormald, in her essay on the James' reputation in England and Scotland, makes clear that James' speech patterns won him no favors in England. For the English elite, to speak properly was of the utmost importance, and James' accent was indicative of his foreignness. James' education growing up included learning multiple languages, including French and Latin. James is reported to have said in his youth that "Thay gar me speik latin ar I coudl speik Scotis". The text of that quote both indicates that James knew he was speaking a dialect, and the spelling (written by his tutor) is indicative of such an accent.

I have not watched the series "Gunpowder" so I can't say if their depiction of a Scottish accent from the sixteenth and seventeenth centuries is correct. I can say that James had a distinct way of speaking compared to Elizabeth or the entrenched English nobility, and that his way of speaking was associated with his Scottishness.