I presume you known the significance of the Bassanos and royal music? They were recorder players ay Elizabeth's court, but I'm sure you know that. I believe Elizabeth Damon/an was the daughter of William Daman (Damon), also a court recorder player. However, I can't find a birthday for your Elizabeth. Damon had a child called Elizabeth whose funeral was 5 April 1572, and we know he and others named subsequent children with the same name (vide Adam DAmon d. 26 Oct 1574, 2nd Adam Damon bap. 3 May 1581).
I'm a musicologist hunting connections for William Damon, the subject of my Master's theses.
Hope there is some fresh info here!
All the best,