First of all thanks for your question. To help your understanding of the Congo Free State develop a bit, there are a few answers I have written previously that address some of the common misconceptions and the history and manner of rule of the territory. To reiterate what /u/holytriplem has said, it was not the Belgians who were viewed as the perpetrators of the genocide in the Congo, but Leopold and his company who were rightly blamed.

To address your original question then, it is also crucial to understand that there were varying schools of thought by 1914 on the colonial efforts of the European imperial powers. You are absolutely right in saying that it was a British led condemnation that saw Leopold lose his private colony, but generalising about the average British person's attitude is incredibly difficult as it was not the official British policy to condemn Leopold's owning of territory. From the late 1890s through the first decade of the 20th century, support for the British empire was high, especially during and after the Boer Wars around the turn of the century. Events such as the Siege/Relief of Mafeking were highly popularised, and jingoistic sentiments were engaged and developed through the advent of what could be called 'modern war reporting' (i.e. news from the battlefields arrived hours or days rather than weeks after the event). The British were proud of their empire (good write up here about the period - non-academic article though so use with caution.) In terms of the actual outbreak of war, the Belgian Congo played little part in determining British response.

Cecil Rhodes had only recently and very violently put down a native uprising in his company's private territory (Southern Rhodesia) and the British had been reluctant to condemn the actions there, but there was a wideheld belief that the British empire was different and 'better' than those of other European powers. The Germans were seen as heavy handed and violent, the French much too close to their colonial subjects, and the Italians as examples of abject failure in running an empire, but most British believed their empire treated all of its subjects fairly, providing them with civilisation and religion and infrastructure and all the other things that Africans 'needed'. Towards the end of the 19th century though a very small, very vocal minority of the British public began putting forward an explicitly anti-colonial agenda and sought to cast Leopold as the rogue of the Congo Atrocities, acting as a despotic megalomaniac would when given absolute power. A report by a British consul, Roger Casement, brought to light in the English speaking world the scale of the suffering in the Congo and others furthered the humanitarian effort and it was this instead upon which the British fell with their full support - they condemned not the holding of the territory by Leopold but his actions towards the peoples within it for being barbaric and unbecoming of a civilised power.

Perhaps the most notable example is Joseph Conrad's fantastic novel, The Heart of Darkness. There is a rather decent article available here which explains the British attitude towards the Congo and the resentment that was directed more so at Leopold, himself an explicit advocate of economic racism against colonial subjects, than at the Belgians as a nationality. This vocal anti-colonial minority campaigned hard and publicly to highlight Casement's report and the testimonies of missionaries working in the EIC as to the scale of the mistreatments and deaths, and spur others to act, specifically the Belgians themselves.

Guy Vanthemsche in his great book Belgium and the Congo, 1885-1980, states that:

Between 1885 and 1900, the Congolese issue flared up at key moments. But at the beginning of the new century, the atmosphere changed as the international campaign against Leopold II increased. This had repercussions in Belgium. Some factions of the Belgian press and, increasingly, public opionion were strongly opposed to the Congo Free State policy. Parliamentarians protested against inhumane practices in Leopold's African state and the Congo became a divisive issue in Belgian domestic politics. (p.41)

It was only in 1906 after the threat of an international conference on the Congo issue that Leopold conceded to allow the Belgian state to annex the colony. The Belgians, although perhaps not as supportive of their colonialism as the British were of their empire, nevertheless held regular exhibitions of Belgian colonialism in the Congo from 1898 to the 1930s. The actions of Leopold were seen by both themselves and the wider world as a different time and the Belgian Colonial Office and Ministry of the Colonies fought hard to portray an extreme contrast of the 'before' and 'after' of Leopold's rule and the annexation of the EIC. Unlike in France or Britain, there was almost no migration from the EIC to Belgium during Leopold's rule and subsequently, Belgian colonialism retained some distance from the general public, that impacted on the public opinion and the way it was shaped by international condemnation. Leopold was very aware of international public opinion and the effect it had on domestic politics in Belgium, going so far as to pay a lawyer to lobby the US Congress with regards to the Congo issue, a plan which spectacularly backfired in December 1906 when the dealings were made public and the US and Britain very quickly pledged complete support in removing the EIC from Leopold's control (see Martin Ewans in European Atrocity, African Catastrophe p.227)

The British in 1908 demanded the abolition of forced labour in the colony and to safeguard the rights and freedoms of the Congolese peoples, a policy supported by the Americans. The British public opinion on this is difficult to ascertain - newspapers were generally in favour of the abolition but it was made repeatedly clear that the problem lay not with the Belgians who were willing to do everything possible to guarantee the future of the people in the EIC, but with Leopold. Leopold's successor, Albert I, made public declarations of intent as to a better future for the Congo (and unlike Leopold actually visited the colony). If newspapers can be seen as a hazy reflection of the opinion of the public they write for, then the atrocities in the Congo were widely condemned.

Unfortunately, making more accurate statements about the public attitude is nigh on impossible and I don't know enough about WWI itself to pass judgement on British attitudes towards Belgium during the outbreak of war, so maybe somebody else could fill in the gaps. I hope this long-winded, rambling answer helps you a little. Obviously, my knowledge is primarily African orientated with the metropoles themselves playing out in the background, but if you have any more questions fire away and I will do my best. The books I referenced are absolutely great reads and probably available on google books for free - have a gander and a read if you want.