The Powers benefited from the cheap cotton the plantation system could provide to their textile mills, so opposition was always more a matter of moral repugnance rather than national interest. While there were sporadic efforts to police the slave trade, these were hampered by jurisdictional questions and the refusal of some States to cooperate with proposed solutions. Since the slave trade was international by nature, it required some form of international enforcement. Mixed Commission Courts were set up to try and punish slavers, but the US refused to allow these courts jurisdiction over American slavers. (We weren't alone in this) Short of war, there wasn't much the Powers could do about that, and they were not prepared to fight a war over slavery.
As for the practice of slavery, most of the Powers had abolished slavery well before the 19th century, although in a rather piecemeal fashion. But the attitude towards slavery in other countries was (and largely remains) that it is a matter for that country to decide, there is no mandate to interfere in the internal affairs of another State.
So what "pressure" might have been brought to bear would have been the efforts of religious and civil organizations deploring the practice and calling for abolition. There was about as much of that as you might expect, and it was about as effective as you might expect. To paraphrase Stalin, the Anti-Slavery Movement had no divisions.
-- Mal