I don't think its being particularly cynical to say if the industries supported by slavery had not been supplanted somewhat (reducing their political power) then slavery would have continued apace. Politicians have proven adept at taking advantage of popular movements when suitable.
Its basic economics. Had the ruling elite deemed it necessary I'm sure Punch could have been commissioned to print all manner of xenophobic cartoons about subhuman slaves and their barbarism in order to justify the practice further.
This isn't to diminish the efforts of the anti-slavery movement or the individuals involved.
*Edit: Also, yes, this is not unique to any country at all. I'm sure there are a myriad of examples of slavery within any of the countries that Britain colonised throughout history.