First off, slavery in the colonies pre-dates "Britain". Britain only properly existed after union of the Scottish and English crowns in 1707.

The first Africans to arrive in the English colonies had been captured slaves from a Spanish ship. The English got most of their early slaves from the Dutch and Portuguese—they did not have to enslave anybody. The earliest slave society in the English Atlantic developed in Barbados during the 1640s borrowing Brazilian methods of sugar production that Dutch merchants introduced to the Caribbean. Winthrop Jordan calls the rise of slavery in the English colonies an “unthinking decision.”

In North America, labor in Virginia tobacco fields was scarce. Labor was mixed early on; indentured servants and slaves worked and lived together. The initial outlay for a servant bound (usually) 5-7 years was lower than purchasing a slave, which made economic sense given the disease environment that fueled high mortality rates in the early years. The choice for slaves increasingly made sense after Bacon’s Rebellion, in which white laborers and slaves joined together in resistance against wealthy planters. Edmund Morgan has famously argued that the expansion of black slavery (1) reduced the importation of white indentured servants and the pressure they created once freed on land supply; and (2) established a racial hierarchy in Virginia that gave poorer whites a stake in the interests of elite whites.

Strictly speaking, racialized slavery was enshrined in a Virginia statute in 1662, which stipulated that the status (slave or free) of a child followed that of its mother. The status of slavery became heritable. Slave codes were developed at the colony level, not by the Lords of Trade and Plantations in England.

Britain did in fact make efforts to limit the expansion of slavery prior to the American Revolution. When Britain gained the Floridas after the Seven Years’ War in the Treaty of Paris (1763), it was stipulated that they should not be slave societies but instead be settled by white Protestants. While there were some attempts to plant white settlers from England, Barbados and Greece, they were slow to occur. Almost immediately, well-to-do planters from Scotland and the Carolinas got enormous grants of land, bought up slaves, hired overseers, and created a slave-based plantation economy.

The question of Virginia wanting the slave trade stopped was not one of humanitarian concern but of economics. After the Seven Years’ War (and all the spending that went with it) ended, the colonies fell into an economic depression. Compounding colonials’ problems was the Currency Act of 1764, which banned colonies from printing paper money. Ending the slave trade or discouraging it through taxes—which were instituted—would prevent the outflow of British specie from the colony. The tax also served to raise the value of slaves already in the colony (part of wealthy Virginians’ asset portfolio, so to speak). It might also be pointed out that taxes levied on slaves discouraging their import fit into broader Patriot strategies of non-importation during the 1760s and 1770s. Some Virginians argued the curtailing of the slave trade made the colony safer by lessening the threat of slave rebellion.

In short, both sides were culpable.

Edit: forgot a word