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
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.
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.
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.
Hope this helps.