Dr. Andrew Zurcher has identified the writer of this letter as Robert Drury, the husband of Anne Bacon Drury. Anne Bacon Drury was the niece of Henry Neville's stepmother, Elizabeth Bacon. The Drurys were major patrons of John Donne, you can read it about it in the book Donne & the Drurys.
I have discovered an extremely interesting letter in the National Archives.
The letter was written to Henry Neville while he was ambassador to France. It mentions both the Earl of Southampton and Humphrey Fludd. Here is the letter, shared with permission of the National Archives. The designation of this letter is PRO 30/50/2/97. Here is the link to the Archives.
Here is a modern spelling transcription, done with the help of John O'Donnell:
Sir, I did not a whit mistake what I writ unto you concerning the letters, for proof whereof Humphrey Fludd, who hath been here with the Commander of Dieppe, assureth me he saw my Lord of Southampton's letter to me, upon a table in your house, laid down amongst many other letters. And for Mr. Secretary's letter, he told me himself, at my coming over hither, he had sent it by post in your packet. But for this I do think it might very well be a compliment, that it pleased Mr. Secretary [Robert Cecil] to use me, and for the other, it might very well be lost not being cared for by anybody. But Sir, those things, are not worth any more speaking of. I will not be jealous it you would willingly do me any wrong, and so will rest.
Your very loving friend to do you service.
London. 6. May. 1600.
(R. Drury)
Here is more information on Humphrey Fludd. He was deposed in the Bellot v. Mountjoy case along with William Shakespeare.
We know that Humphrey Fludd was a professional courier. It is mentioned in Winwood's Memorials, here Ralph Winwood specifically mentions a letter sent via Fludd to Henry Neville.
The Shakespeare Circle: An Alternative Biography by Paul Edmonson and Stanley Wells, cites a secondary source, A Biographical Dictionary of English Court Musicians:
"Between 1608 and 1618 he was paid numerous times for carrying official letters to and from France."
Update: Here is a reference to Robert Drury:
