I think it's complex, public opinion really started to tilt heavily in favour of the civil rights movement when they had big peaceful marches in segregated southern cities and were attacked by the police, they were seen as victims of the southern police forces and local elites. Before then there seems to have be scepticism and criticism of disruptions etc. I think that tactics do matter and other groups who employ disruption etc like feminists and environmentalists have been shown to get a negative image because of this, especially when people don't 100% agree with or understand their goals.

Also much more importantly imo King and some of the other civil rights leaders and committees actively tried to involve universal concepts and campaigned for white people in similar situations. When he was shot King was supporting a strike by sanitation workers as part of the 'poor people's campaign', black workers were a large part of those who would benefit but this was a universal concept that white and other groups benefited from also, this seems to be how you build broad based support.

I kind of feel like the kind of exclusive black nationalism that blacklivesmatter represents is not particularly applicable to the UK and even in the US it is more complicated and less overtly racialised than presented. There is a overrepresentation of black men (and other ethnic minority groups) in searches, arrests and prison sentences in the UK, but imo this is as much about where you live and what you're wearing than what colour your skin is. I bet the differences would disappear if you looked at white groups living in similar circumstances on council estates or deprived urban areas, with similar 'informal economies' and gangs like parts of Salford etc. Something similar is true of the US but the concentration of urban poverty is much more pronounced and police have been brought in to 'manage' areas where there is high poverty. There are lots of cities where there is a high rate and concentration of black poverty and the stress policing that seems to go along with it, but in towns and cities were there is a high rate of white poverty in some parts of Appalachia and the mid, west the treatment of those white people isn't really all that different and there are even similar stop and search policies. The relatively high sentences that black men get (or used to get) in New York is similar to the law allowing police to arrest people holding even small quantities of various combinations of household ingredients that you can make meth from. The only difference is that white people are less likely to live in areas of high poverty, high unemployment and the kind of 'informal economy' that goes along with it.

I kind of don't really want that kind of sentiment to come over here and increasing the social divide between communities that aren't really divided by policing so much as class and where you live. An important factor is also that there is nothing even approaching the US rate of incarceration or police killings of unnamed people in the UK.

I think that the divisiveness and lack of alliance building is reflected in surveys, according to the opinion surveys I've seen Blacklivesmatter doesn't seem to have changed white opinions very much at all, they remain kind of ambivalent.