Liverpool midfielder Henderson won the men's award ahead of second-placed Raheem Sterling and third-placed Harry Kane.
Bronze finished on top of the fans' poll in the women's category, with 2018 winner Ellen White coming second and Beth Mead in third.
It is the second time Lyon star Bronze has claimed the England honour, having previously won it in 2015.
Henderson made seven England appearances last year, passing the 50-cap mark during a 5-1 win in Montenegro.
He also helped Liverpool win the Champions League, European Super Cup and club World Cup.
Bronze, 28, has won 81 caps and was a key figure when the Lionesses lifted the SheBelieves Cup for the first time and reached the World Cup semi-finals.
She played in 18 of England's 20 games last year, highlighted by a superb goal in the World Cup quarter-final victory over Norway.
Bronze was also named 2019 UEFA women's player of the year, having helped Lyon win four trophies, including the women's Champions League.