The Calendar of State Papers: Colonial Series, vol 2 East Indies, China and Japan, 1513-1616, p. 321 does mention John Mildenhall (written as Midnall) in a letter from Thos. Keridge to the East India Company, dated 20 Sept 1614. It states John Midnall has died leaving his goods to his two illegitimate children in Persia by an Indian woman and naming a Frenchman as his executor. The Frenchman has promised to marry the daughter and bring up the son, which would suggest that the son at least and probably both children are minors.
The fact that the children are illegitimate would seem to make more sense, given that his wife back in England doesn't seem to have died until 1602. Also the fact that he has left them some goods (though it appears they weren't his to leave) and not his entire estate also sounds more reasonable.
Although there are other entries in this volume for John Midnall or Mildnall see index on p. 530 I couldn't see that any others name the children, though I didn't check them all (note these are entry numbers not page numbers).
