I think the correct term you're referring to is 'indentured servitude'?. In any case, the new institution of 'apprenticeship' was not chattel slavery - sure, it was wage slavery; but they are not the same things.Source: The Guardian
Technically, & only technically, yes Britain outlawed Slavery at the stroke of midnight on 1 August 1834, & the enslaved were freed from the "legal category of slavery" – and instantly plunged into a new institution, called “apprenticeship”.
What the so called ex-slaves had to endure, sanctioned by the 1834 British Law, to many was far worse than slavery ever was.
So, your typed question, as relating to Britain, though technically valid, was followed by nothing less than what many described as merely an extension of slavery by the British, technically called “apprenticeship”, sanctioned by the British Government, which legally flourished until 1838.
The 13th Amendment to the US Constitution, ratified by the American People in 1865, effectively freed all slaves within the United States. It forever abolished slavery as an institution in all the U.S. States & territories. In addition to banning slavery, the 13th Amendment outlawed the practice of involuntary servitude.
Second, the abolition of slavery in America was replaced by Jim Crow, so even though it was constitutionally banned, it didn't prevent discrimination against blacks.