As far as I know, Judaism has no tradition of removing a head covering / hat during prayer. Therefore, the Christian tradition started sometime after the split and it is not an inherited tradition. Where does this Christian tradition come from? Is it just a western or US tradition for males?
One of my churches used this passage when requesting all men remove their hats during times of prayer:
1 Cor. 11:4
Every man praying or prophesying, having his head covered, dishonoureth his head.
In certain cultures, it's possible that the hat is removed and head is bowed as a sign of respect and not necessarily from this verse.
This tradition was apparently begun by Paul as a sign of respect for Christ.
But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God. Every man praying or prophesying, having his head covered, dishonoureth his head.
1Cor 11:3-4 - KJV
Paul also said that a woman should never pay uncovered and went on to explain to the Church at Corinth his reasoning for these signs of respect.
But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head: for that is even all one as if she were shaven. For if the woman be not covered, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered. For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man.
1Cor 11:5-10 - KJV
For the man is not of the woman; but the woman of the man. Neither was the man created for the woman; but the woman for the man. For this cause ought the woman to have power on her head because of the angels.
1Cor 11:8 - KJV
As far as I can tell Paul was not setting this out as a requirement, only pointing out that not praying this way was not giving proper honor to either Christ or God. As far as the woman goes praying uncovered was dishonoring all three God, Christ and man.