Idealism and the post-Idealistic atheistic nihilism of Schopenhauer and Nietzsche did not make their way over to Britain until later in the century (or, in the case of Nietzsche, not until the next), but British atheism in the early to mid-nineteenth century was principally shaped by the importation of eighteenth century French atheism. As the philosopher David Berman has established, the first avowedly atheistic book in Britain was the Answer to Dr. Priestley's letters to a philosophical unbeliever (London, 1782). Although the identity of the author of this text is unclear, it has been attributed to a Liverpool physician by the name of Matthew Turner.
However, two of the most important figures for understanding nineteenth century British atheism are the social reformer Robert Owen (1771-1858), and Thomas Paine (1737-1809). Both of the latter were deists, not atheists, although Paine represented a radicalised form of deism which was disseminated beyond the privileged classes to the lower classes also. Paine's Age of Reason, published in 1793-5, was the most influential deistic work of the Anglo-American Enlightenment. In the hands of working men such as Richard Carlile (1790-1843), G.H.Holyoake (1817-1906) and later Charles Bradlaugh (1833-1891), figures largely forgotten today, the attenuated deism of authors such as Owen and (especially) Paine was transformed into atheism, and British atheism, as in France several decades before, became not merely a high brow preoccupation but a political movement which was spreading rapidly among the middle and lower classes. The first campaigning British atheists were at first deists, but shifted (generally through enduring some sort of persecution) to atheism (Carlile, for example, seems to have converted from deism to atheism during a period spent in prison for publishing Paine's Age of Reason). In 1866 Bradlaugh co-founded the National Secular Society, to this day one of the chief lobbying groups for atheists in the UK including members such as Richard Dawkins and Christopher Hitchens.
In addition to this there were new influences from post-revolutionary France - socialism and positivism. The British utilitarian Jeremy Bentham (1748-1832) (thought to have been one of the two anonymous co-authors of the Analysis of the influence of natural religion on the temporal happiness of mankind (London, 1822), one of the most powerful British atheist tracts of the first half of the nineteenth century) laid the basis of English socialism, but Comtean positivism also played a role, in particular through its influence on John Stuart Mill (1806-1873) (Mill was not himself an atheist but an attenuated deist, as his own opinions on religion published after his death made clear; his ideas, however, would galvanise a generation of philosophical atheists such as Holyoake and Bradlaugh).
A further influence that must be mentioned here - although not directly atheistic - is the publication of Charles Darwin's Origin of the Species in 1859. Darwin's evolutionary theory, although widely seen as undermining the traditional design argument for religious belief, was not itself atheist, nor was Darwin himself atheistic by inclination (he vacillated between deism and agnosticism) - but his theory of the origin of life was elaborated by his philosophical successors into a materialistic system and quickly extended to the social sciences. In Herbert Spencer (1820-1903) this philosophical appropriation of the scientific theory of evolution would ultimately encourage a reductive Social Darwinist understanding of social life, including religion.
A History of Atheism in Britain: From Hobbes to Russell. London: Croom Helm, 1988.