Lose not an hour can be found before Nelson’s time (1758 – 1805), such as in The practical works of the late reverend and pious Mr. Richard Baxter from 1707:
And doubtless this is the best way to redeem time, and fee that we lose not an Hour, when we spend it only on Necessary things : And I think it is the way to be most Profitable to others, tho* not always to be most Pleasing and Applauded; ...
However, as old-fashioned as “lose not a” sounds to our ears, it’s probably just that: old-fashioned. I don’t think it would have sounded odd in the 18th or 19th centuries.
This Ngram chart suggests “lose not a” was more popular than “don’t lose a” (and “do not lose a”) during the 19th century, plus a bit each side:
(Note: Google processes “don’t lose a” as “do not lose a”.)
English has changed so that we now usually say “do not lose” or “don’t lose” rather than “lose not” (and likewise with other verbs). Remnants of the old way can still be found, for example in poetry and this from the King James Bible (1611):
Then said Jesus, Father, forgive them; for they know not what they do.
This sounds archaic (although now idiomatic) to our ears, but would have sounded normal in the 17th century. If we compare modern translations from the New International Version (2011) and New Living Translation (2007):
Jesus said, “Father, forgive them, for they do not know what they are doing.”
Jesus said, “Father, forgive them, for they don’t know what they are doing.”