I don't agree with the original statement. Most firms I know are have staff from a variety of different backgrounds.
Ultimately the majority of people in UK firms are likely to be white, because the majority of people in the UK are white. I don't think racism is at play. Most firms genuinely hire based on academics and experience.
More than many other industries, academic achievement is highly regarded in consulting. Without wanting to open a can of worms, statistically speaking, the groups you say are under-represented don't generally perform highly in academia. But that is changing all the time, and I expect corporate diversity to continue improving over the next few years.
I'd be more concerned with the disparity in terms of the gender mix in consulting. That certainly doesn't reflect the UK's population...