This might be more complicated, but could be less complicated. One of those pieces of information says that his father owned several plantations in Antigua and served on Antigua's Council, etc. There is no Antigua in Virginia, so perhaps (and likely) the reference is to the island of Antigua, a British possession in the Caribbean. In that case, he probably grew up on Antigua, and not in Virginia. It also easily explains why he joined the British Navy and not the Revolutionary army (only one ship in the "Navy"
), though it is possible that his family were Loyalists in the colonies.
By coincidence, Lord Nelson came to Antigua in 1784 to develop Naval facilities on the island:
http://www.antigua-barbuda.org/Aghis01.htm. The American Revolution officially ended on January 14, 1784, but perhaps Nelson picked up this young sailor when he left Antigua?
Assuming his family lived first in Virginia, then moved to Antigua, and not the reverse, I suggest extending the search to Antigua. Finding out the exact place of his birth in Virginia will probably not be possible with such scant information, especially if the family moved to Antigua afterwards.