If there was a Henry Waring Lee, which isn't supported by any primary source records but only by speculation in undocumented family tree's, it's reasonable to question that he has been identified with the wrong father: His father would not have named two sons Henry, one legitimate and one illegitimate. It would have been an insult to his legitimate son.
It’s supported.
ttps://www.stirnet.com/genie/data/british/ff/fletcher05.php
A. William Fletcher of Campsall, Yorkshire (b 1532, bur 05.04.1590) m. Elizabeth Wentworth (dau of John Wentworth of North Elmsall by Anne, dau of Sir Brian Hastings of Fenwick)
ii. Elizabeth Fletcher m. Henry Lee alias Waring (natural son of Robert of Hatfield, the brother of Sir Henry)