Wrey was the son of Sir Chichester Wrey, second baronet, by Anne, widow of Lionel Cranfield, Earl of Middlesex, and daughter and coheiress of Edward Bourchier, Fourth Earl of Bath (d. 1636).
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Amongst his possessions at his death was the Illyrian armorial with the arms of the families and surnames of Albania, Bosnia and Herzegovina, Bulgaria, Croatia, Dalmatia, Macedonia, Montenengro, Serbia and Slavonia from the Armorial of Stanislas Rubčić, King of Arms to Tsar Stephen Uroš IV Dušan of Serbia.