I like the Simon Lake theory but he was a bit late
The first definitely credible submarine was built by Dutchman Cornelius Drebbel in 1620, he may well have drawn on the plans of William Bourne from 1578.
Incidentally, when checking the dates for the reign of James I (and VI for any hailing from north of the border), I came across a Wikipedia entry (I know, I know) which managed not to mention one of the defining facts of his reign, that he was a protestant! And we wonder why Halloween is replacing Guy Fawkes/Bonfire night. James I (and VI) reigned from 1603-1625 in case anyone cares.
The first credible military submarine is less clear. By the early 18th century various designs were made and a few even built. The colonial traitor David Bushell designed a one man submarine and the rebels even claimed to have attacked a British ship with it in 1776, only problem is the British have no record of any submarine attack. Maybe a bit of American propaganda?
Another American, Robert Fulton, designed a submarine for the French which did prove capable in tests, but was never tried in anger. Now, however, we are in 1800 and the Napoleonic wars.
The first submarine NOT powered by human power was French again, but you have to wait until 1863 for this. It was absolutely useless too!
Surely Hitler evaded the assassination attempt(s) and escaped to South America.....in a submarine!!!!!!!! I think that the documentary you mention may have been shown on the history channel, where all the best and most accurate historical research is turned into compelling and engaging programmes for our delectation. Even the history channel does get the odd fact right.....errrr.....I mean wrong
Cheers
Steve