This is a good question but I see a pretty big misconception, the Bible is very much (overwhelmingly even) in support of slavery and Christianity played a huge part in justifying slavery to Southern slave owners.

Well the main of it is that to American, Evangelical, Christians (ie most Americans at this point), the argument for slavery is much, much stronger than that of the abolitionist. The main argument used is that of "The Curse of Ham" in the Old Testament (Genesis 9-11) after the Great Flood, Noah gets drunk on wine and passes out in his tent, one of his sons Ham, sees his nakedness and tells his brothers, Shem and Japheth. When Noah awakens he is outraged and curses Ham.

“Cursed be Canaan! The lowest of slaves will he be to his brothers.” He also said,“Praise be to the Lord, the God of Shem! May Canaan be the slave of Shem. May God extend Japheth’s territory; may Japheth live in the tents of Shem, and may Canaan be the slave of Japheth.

In this passage Canaan/Ham (The African) is cursed to be the slave of Shem (The Asiatic) and Japheth (The Caucasian). As time went on, the black race was barbarized further by the inclusion of the Tower of Babel to their racial history in which the character Nimrod (grandson of Ham) is blamed for the confusion of the tongues and Gods further separation of the races on the Earth. So by using Biblical justification, they dehumanize the African race and thus make it easier for slavers to morally condone slavery. In addition to this, Biblical scholars of the 19th century point to the Many instances in the Old Testament (beyond the Curse of Ham) to argue that the Bible itself gives support to the institution of slavery. There are several instances in Leviticus that outline slave laws but do not speak against it. One good example is

Leviticus 25:44-46

As for the male and female slaves whom you may have, it is from the nations around you that you may acquire male and female slaves. You may also acquire them from among the aliens residing with you, and form their families that are with you, who have been born in your land; and they may be your property. You may keep them as a possession for your children after you, for them to inherit as property.

In terms of the New Testament, Jesus himself says nothing but it is outlined several times throughout, never in terms of abolition but rather that slaves should obey their masters yet it also indicates that there is a little bit of leeway for Christianized slaves.

1 Timothy 6:1

Let all who are under the yoke of slavery regard their masters as worthy of all honor, so that the name of God and the teaching may not be blasphemed.

Now for Americans who want to use the Bible for Abolitionary purposes, they have to contend with nearly a hundred Old and New Testament verses that allow and condone slavery (If you want a list, I can send it to you.) So this very strong Biblical defense of slavery is basically unpenetrable without going through some serious theological loops. Slavery doesn't end due to Biblical thought and analysis, it wouldn't have happened that way without a serious overturning of mainstream American religious thought, the argument for which theological argument about slavery is actually correct is solved by the Reverends Grant and Sherman from 1861 to 1865.

In regards to the Church of England, that's out of my wheelhouse but i'd be very interested in looking into that era and community.

Sources:

Noah's Curse: The Biblical Justification for American Slavery, Stephen Haynes

Politics and Ideology in the Age of the Civil War Eric Foner

I was going to actually write a long response, but the one from /u/Dubstripsquads was fantastic, so I don't think there is much else of me to write. However, I will say, that as he pointed out, the bible is actually quite in favor of slavery, and by the time Paul started writing his letters (around 50 to 55 CE), it became a major topic, one in which he constantly encouraged slaves to obey their master's and serve them as they would serve Jesus himself.

In early Christian History, slavery was something that was practiced, both by the newly converting Hebrews, and by the Romans and Greeks who were making up the majority of the new coverts to this religion. Despite the historical Jesus' constant messages about love and fairness, early Christians had no problems looking at Old Testament verses (like the one below) and also the more contemporary letters (like Paul's) to justify their end-game.

If you buy a Hebrew slave, he is to serve for only six years. Set him free in the seventh year, and he will owe you nothing for his freedom. If he was single when he became your slave and then married afterward, only he will go free in the seventh year. But if he was married before he became a slave, then his wife will be freed with him. If his master gave him a wife while he was a slave, and they had sons or daughters, then the man will be free in the seventh year, but his wife and children will still belong to his master. But the slave may plainly declare, 'I love my master, my wife, and my children. I would rather not go free.' If he does this, his master must present him before God. Then his master must take him to the door and publicly pierce his ear with an awl. After that, the slave will belong to his master forever. (Exodus 21:2-6 NLT)

and

Slaves, obey your earthly masters with deep respect and fear. Serve them sincerely as you would serve Christ. (Ephesians 6:5 NLT)

Now I am an early Christian historian, so I can't speculate too much on the part in played on the civil war, but I have dedicated major portions of my adult life to studying the bible, especially the New Testament, so I figured I'd try to help.

Good question, let me know if you have any more.