Does the practice of brides wearing veils have its origin in either the Bible or Christian tradition? If so, where exactly did it originate and what is it supposed to symbolize?
Here's a basic explaination I found online. It gives a simple, streamlined answer to at least part of your question.
It would seem the veiled bride (Temple) is from the Word itself!
In John 3:28,29 John the Baptist says: “Ye yourselves bear me witness, that I said, I am not the Christ, but that I am sent before him. He that hath the bride is the bridegroom”
When Christ’s work on the cross was “finished” the “temple veil was rent” (Matthew 27:51, Mark 15:38, Luke 23:45) and the “Holy of Holies” in the Temple was unveiled.
In Ephesians 5:25 “for the husband is the head of the wife, even as Christ is the head of the church: and he is the savior of the body. Therefore as the church is subject unto Christ, so [let] the wives [be] to their own husbands in every thing. Husbands, love your wives, even as Christ also loved the church, and gave himself for it”
One could and if qualified one should study the nuances between “veil” and “vail”. However, in my brief non-credentialed study “vail” is an archaic rendering of “veil”. In both cases a veil/vail is a covering. The vail between the Holy place and the Holy of Holies was also used as a covering over the arc of the covenant when in transit.
Veiling and unveiling are significant considerations as in 2 Corinthians 3:13-18: “And not as Moses, [which] put a vail over his face, that the children of Israel could not stedfastly look to the end of that which is abolished: But their minds were blinded: for until this day remaineth the same vail untaken away in the reading of the old testament; which [vail] is done away in Christ. But even unto this day, when Moses is read, the vail is upon their heart. Nevertheless when it shall turn to the Lord, the vail shall be taken away. Now the Lord is that Spirit: and where the Spirit of the Lord [is], there [is] liberty. But we all, with open face beholding as in a glass the glory of the Lord, are changed into the same image from glory to glory, [even] as by the Spirit of the Lord.”
The Law veiled the heart but Christ unveiled believers (the Church) so “with open face we can behold the glory of the Lord”.
According to John the Baptist Jesus is the Bridegroom
According to the Gospels when Christ “gave up the ghost” the temple (Church) veil was torn in half.
According to Paul there is a real parallel between Christ and the Church and husbands and wives.
According to John in Revelation the Bride is the lambs (Christ’s) wife (Ephesians 5:27) “That he (Christ) might present it to himself a glorious church, not having spot, or wrinkle, or any such thing; but that it should be holy and without blemish.”
I have nothing outside of Scripture to make the claims that I have made, so I do not know if and when the Church adopted bridal veiling but I believe I can speak with confidence that the above verses would be the best reason for it to do so!