The treaty of Ghent, which ended the war of 1812, stated that both parties would work to end the international slave trade. There were also disagreements upon the issues of impressment, which the two parties had to have subsequent agreements (sorry no source). The sticking point was that the British had to return US citizens that were impressed into naval service. The British argued that by US laws the US slaves that were impressed were not citizens but property and there for should remain in British Naval service. There was little fear in the British colonies that they would end slavery outside of the UK. The slave system profited plantation owners in the Caribbean of course but it was an enormous wind fall for the UK. Raw materials would come in from the colonies the refining or milling or whatever would be done in the UK and then sold globally. The British military was an ever present partner with the colonies to maintain the status quo, including slavery. One of the major turning points was the 1831 slave rebellion in Jamaica. The opinion of colonial slavery dramatically shifted in the UK so much so that British slavery was abolished a few years later.

I would argue that it isn't until this time period that whites living in British colonies would even consider that British slavery in the colonies would be abolished and what domestic policies they chose to enact in the homeland had little bearing on life in the colonies. 1776 was 50 years before this. The full industrial might of the slave system in the Western Hemisphere was still 20 years or more in the future. I think abolition within the UK had little or more likely, no bearing on the build up to the US Revolution.

Not related. The British abolitionist movement was not a thing in the 1770s. The revolution was about taxes, troop quartering, representative government, and the divergence of colonial and metropolitan cultures