Even prior to Apartheid being officially established there was opposition to the practice of segregation in South Africa. In 1948, the year the National Party took power, and often seen as the beginning date of Apartheid, Alan Paton released a damning book on the treatment of the native population with 'Cry the Beloved Country'. The book itself explores the opposition to Apartheid by both the Anglican clergy, and the liberal whites themselves. It should also be mentioned that against the National Party in that election stood Jan Smuts and the United Party, who were committed to eventual racial integration and opposed to the National Party's plans for Apartheid.
During the Second World War, the Springbok Legion was found within the South African Defence Forces, which was avowedly anti-fascist and anti-racist. Prominent Jewish members of this group such as Lionel Bernstein and Joe Slovo would even go onto join Umkhonto we Sizwe, the armed wing of the African National Congress. The Springbok Legion would transform into the Torch Commando within time, with torch-lit marches led by Adolph ''Sailor'' Malan, an Afrikaner WW2 fighter ace, but also an unashamed liberal who opposed Apartheid. These marches would attract thousands until the Apartheid government put a stop to them.
Other movements such as the Black Sash movement, which is often forgotten was led by housewives, mostly English speaking, who were against Apartheid. As the war in Angola and Namibia dragged on, there would be considerable internal opposition to conscription with the End Conscription campaign. In scenes echoing the internal opposition to the Vietnam War, young men would protest, feign insanity, or escape to abroad to avoid military service.
In parliament, during Apartheid there was always liberal opposition standing against the National Party, who would use their parliamentary privileges to expose the crimes of the regime. Prominent leaders were Harry Schwarz and Helen Suzman. It should be noted that these parties never got over 20% of the vote. Other whites would use their position of privledge to advance the cause of truth and justice, i.e. Helen Zilleman, a journalist, would expose the police's murder of Steven Biko, and Peter Hain, of Anglo-African descent, organised an extensive anti-Apartheid campaign and movement from the United Kingdom after his family were essentially exiled from South Africa due to their political opposition to Apartheid.
White opposition to Apartheid has existed for as long as the policy itself. Many whites, of Jewish, Afrikaner, British and other descent spoke out against Apartheid. Some did so at the ballot box, others organised movements with the white community itself to oppose Apartheid, and others directly took arms and engaged in violence resistance. Despite their varying different degrees of involvement, they were all united in their effort to see Apartheid dismantled. As for the influence of the African American civil rights movement on white opposition to Apartheid, the literature is spare on it, so it is difficult to discern if there was considerable influence. However, by the early 1960s, South Africa was effectively facing an armed stand-off, whereas despite a few atrocities the American civil rights movement remained largely peaceful. The African American civil rights movement had a huge impact on how the ANC mobilised and organised against Apartheid, taking similar steps of bus boycotts and sit-ins. This unfortunately was cut short following the Sharpeville Massacre in 1960, where 69 protestors were shot dead, which was the direct catalyst the the ANC starting the armed campaign against Apartheid.
Sources:
A History of South Africa - Leonard Thompson
Cry the Beloved Country - Alan Paton (Fiction but a brilliant account of South Africa during the 1940s)
Long Walk to Freedom - Nelson Mandela
...
Edit: I just want to add more context so this doesn't appear as apologist. The majority of White South African benefited Apartheid, and supported it either directly or indirectly. Although a not insignificant number were opposed to Apartheid, they were largely apathetic in their opposition. Addressing the specific issue of British South Africans, they were less supportive of Apartheid in general compared to Afrikaners, but a significant number still voted for the National Party particularly as they started to appeal more to such voters during the 1970s. Not to detract from the heroic tales of sacrifice and resistance from some White South Africans, but it must be viewed in the correct context that I feel my original answer did not properly address.
And likewise to add further to that context, after the series of national emergencies and near state of civil war during the 1980s and early 1990s, 68% of White South Africans voted to continue with negotiations and to write a new constitution to end white minority rule in 1992.
Short answer. Yes, there was at least 1 white South African of British descent who spoke out against apartheid. It mainly occurred towards the end of Apartheid in the last 10 years of it when SA was collapsing due to global economic sanctions and losing of wars that many whites perceived as ending in total disaster.
African American civil rights movement information was heavily suppressed in South Africa and the colonized parts of Africa. The British and French went so far as to outright ban the writings and books of African American major publications like that of Marcus Garvey. Many white south africans did not hear or know of global or world events as the government had heavily censored the media, it was a fascist dictatorship, where like most dictatorship only a limited amount of approved people could vote who typically constitute around 10-15% of the population. In other countries it would usually mean being a member of the governing party or the baa'th party or the CCP or something like that, in SA the qualifier was being of white race.
Probably the largest influence of African Americans on white South Africans was in the sense that they voted heavily Democrat, and the Black Democrats in the 1980s had pushed heavily to sanction the racist apartheid dictatorship; as the Dems controlled the House in the Congress of the USA had pushed to sanction South Africa's apartheid dictatorship that influence had crippled the ability of many white South Africans to conduct business globally.
American sanctions are crippling because they cut you off from the global trade. US sanctions don't just mean you can't do business with USA, it means you can't do business with anyone USA does business with indirectly, so you quickly find yourself being unable to export much except to handful of countries who will look the other way for you (Japan and UK to some point for South Africa)
The vast and overwhelming majority of whites in South Africa supported Apartheid. The only time that changed was at the end of the frontline wars, when SA last ally, USA pulled support because the Democrat congress overrided Reagans pro-Apartheid policy and sanctioned them. Combined with a series of losses, (basically losing every war from the Congo on down) known as the frontline wars, having lost the entire territory of Namibia and being encircled by the ANC on all sides.
Suddenly many white people had a change or heart after taking heavy causalities in this series of wars lasting decades and realizing they were going to lose a war in South Africa so decided to vote for democracy and to end apartheid. So eventually many white Brits (and Dutch) South Africans came around.
Grant it, they did this mainly so they could try to hold on to all the resources they had stolen from local Africans, rather than any real opposition to Apartheid policy.
In the UK, there was pressure as well and with the fall of communism, there was very little left to justify in propping up the Apartheid fascist dictatorship under the guise of Capitalism and stopping the spread of communism.
I have gotten a tad ahead of myself here. But to give background information, there existed a belief among many right wing people in the Anglo world that the ANC was communist or would turn South Africa communist if they took over and democracy was established in SA, therefore countries like USA, UK, Canada, etc backed the apartheid regime under this false claim. By the late 1980s communism, was collapsing, specifically in Europe with the fall of the USSR. So the justification for backing a brutal apartheid dictatorship had completely fallen through.
Very few white South Africans opposed apartheid because they grew very wealthy off of it. Whites in South Africa and Namibia on a whole were wealthier and had higher standards of living than their counter parts in Europe or the USA. They had a black workforce who they basically didn't have to pay in conditions not much better than slavery.
Only by the 1980s when SA was under heavy sanction and the Africans had mobilized to render apartheid ineffective through a combination of protest, military action, international pressure, sanctions and so on, did SOME white South Africans come to the realization that Apartheid would fail so they sought to keep their wealth and try to switch to a democracy that allowed them to essentially keep all their stolen goodies.
Most of the whites who did oppose apartheid prior to it become a complete disaster were jews like Helen Joseph, but Jonty Driver was British descendant and an anti-apartheid activist. I can find no examples of a white south african directly influenced in the way your question describes who opposed apartheid on that basis.
White South Africans who opposed apartheid, opposed it for a variety of reason ranging from it was a human rights abuse, bad for business, it was horrible for the country economy once all the sanctions poured in, they were about to lose a war on their home turf and end up dead, fear of MK -the armed wing of the ANC who carried out attacks in the white areas of SA, etcetera. It was not really a moral decision for most white south africans, in fact most white south africans you'll encounter will typically still try to defend how apartheid "could have worked if...."
Sadly, white South Africans never had the kind of moral or spiritual awakening you saw in USA with the hippies/counter culture in USA and a general acceptance of racism as bad/wrong morally reprehensible. There was no JFK or LBJ equivalent who tried to explain to the nation why racism was wrong. It was merely a practical decision of we are surrounded on all sides, lost a series of wars, are heavily sanctioned, have a failing economy and will probably all die and are getting shot up and can no longer reasonably defend from the increasingly powerful MK wing of the ANC and its regional allies. So a negotiated peace is preferable to death.