Although Frederick Douglass' was well-received in England and attracted many people to meetings, there is little evidence to suggest that his influence had a significant impact on British policy, at least compared to other issues at the time. Nor does there appear to be any evidence that he met with any political leaders.
There is a website called Frederick Douglass in Britain and Ireland. It has pages on his visits to Britain in 1845-7, 1859-60 and 1887 but nowhere as far as I can see does it state that he met any British political leaders, though he did hear William Gladstone speak in 1887.
Among the people Douglass did meet were, most obviously, abolitionists. They not only provided accommodation at times but also arranged lectures. He met the Danish author Hans Christian Andersen, the English authors William Howitt and Mary Howitt, the Irish politician Daniel O’Connell (with whom he shared a platform and also dined), the Mayor of Cork Richard Dowden and the prominent abolitionists Thomas Clarkson and George Thompson.
Douglass was a fine speaker and attracted large crowds (there's a map here of where he lectured) but his impact on British politics was limited according to this assessment by David Turley in The Culture of English Anti-Slavery (cited by Frederick Douglass in Britain and Ireland):
Despite Frederick Douglass’s presence, anti-slavery remained a
“tradition” rather than an activist policy. As a result, some American
abolitionists grew disillusioned with the British public, and while
certain abolitionists such as Douglass could create fanfare, this had
more of a short-term impact on society.
The Cambridge Companion to Frederick Douglass and Frederick Douglass and
the Atlantic World also make no mention of his meeting any British political leaders. The former, though, has a slightly different view to Turley on his influence. Referring to his first visit,
He spent almost two years in England, Ireland, and Scotland, speaking
to ever-larger audiences about the nature of American slavery and
gaining thousands more converts to the cause.
Douglass also gained much publicity and support when, on booking a return ticket on a Cunard ship in 1847, he was denied a 1st class berth on account of his colour. This prompted widespread newspaper coverage, including a Times editorial which began:
The tyranny complained of in a letter signed “FREDERICK DOUGLASS”,
which appeared in our paper of Tuesday, ought not to be allowed to
pass in this country without some public expression of disapprobation
and disgust at a proceeding wholly repugnant to our English notions of
justice and humanity.
Frederick Douglass in 1847.
The editorial went on to describe Cunard's prejudice as "ignorant and contemptible". Samuel Cunard, the shipping line's founder, had to issue a public apology - quite an event for the time. Clearly, Douglass had made an impact during his stay.
This, and the thousands of new recruits to the abolitionist cause, may have had some influence on British MPs and hence the government but it is hard to prove this either way. Also, there is some debate among academics as to what extent the British electorate supported the Union cause. However, on his second visit in 1859-60,
Douglass did not have much success in British society. Was this
because the British public were tired of American slavery? Or had
society become more inherently racist? Or was it simply because
Douglass failed to create controversy as he had done on his previous
visit? Arguably, all of these factors had a part to play, and as a
result of this, Douglass remained in the industrial north of England
and Scotland, where he had been popular during his previous visit.
Nonetheless, during the Civil War, Douglass' speeches were widely reported in Britain. His 1862 appeal to the British to resist recognizing "the independence of the so-called Confederate States of America" and his connections with abolitionist friends who could get his letters and views printed in British newspapers undoubtedly had some impact, but just how much is hard to gauge.
Of greater importance, though, were economic and diplomatic considerations. The British Prime Minister, Lord Palmerston, although personally sympathetic to the South, could not ignore Britain's reliance (50%) on American corn imports. Other major issues meant Palmerston was concerned about
Europe and Bismarck, feared for the
vulnerability of Canada, and desired a stable western hemisphere free
of possible future Confederate adventurism.
Although he did not meeting any prominent political figures in Britain, Frederick Douglass did meet a number of prominent figures in the US, including two presidents.
In 1850 in New York, Douglass attended Fugitive Slave Law Convention with prominent abolitionists Gerrit Smith (who ran for president three times) and Theodore Dwight Weld.
Fugitive Slave Law Convention, Cazenovia, New York. Frederick Douglass is seated at the end of the table, Gerrit Smith is standing immediately behind and slightly to his left.
He met President Lincoln on at least two occasions (in 1863 and 1864). Later, he spoke to audiences which, also on at least two occasions, included President Grant and his cabinet.