I actually don't think that the UK is institutionally racist, at least in the very specific senses of both words; our bodies and structures aren't set up in a manner that specifically excludes those outside the "ruling" race, and you're not automatically more privileged if you're a member of that race.
This latter point is relatively easy to prove. If I were to type the word "chav", your mental image right now will be a scrawny white kid in recognisably branded sportswear. Chavs aren't exactly socially mobile and getting into positions of power - they're white and underprivileged.
What we are is institutionally classist - the issue isn't so much black and white as it is poverty. For decades we've put immigrant afro-caribbean families into the same socialised housing on sink estates that produce the chavs, where the guys with the most money, power, and influence are the people who exploit others and do the most crime. Those that do get put away come out better at doing crime, and the kids look up to them because they have lots of nice stuff and cars, and they cut anyone who crosses them, and it seems cool. They grow up to be them - they don't have to care about school (so they get kicked out and get no qualifications, making it impossible to get out of the cycle) because it's not a necessary aspect of that life. Lather, rinse, repeat, get stabbed to death somewhere between 16 and 35 (after reproducing, so your kids want to get the same power/money/influence and kill the guys that killed you).
Meanwhile almost all of the middle class is white, and ABC1 jobs are white or sub-continent. That means the kids who do well in school and who get to go to private schools or grammar schools are white, Indian, and Pakistani, and they go on to get ABC1 jobs and do the same for their kids - I went to a private school in the north, and of the 175 kids in our year we had 22 Indian/Pakistani/Sri Lankan kids, two east Asians, and literally zero Afro-Caribbeans - every other kid was white. By law of averages it should have been 150 white (140 white British), 12 Asian, 7 black, and six from elsewhere, and yes, almost all of the sub-continent kids are doctors now, even Kishan who had the intellectual capacity of a tangerine and the moral compass of a hyena. I worked at a private school in Cambridge in 2007/8 and there were no non-white kids at all.
It affects more white people than black people in numerical terms, but as a proportion it affects far more of the black population than the white population, which makes it harder for the average black person to cross class boundaries than the average white person. That looks like instutional racism, but it's not because they're black, but because they're born into the poverty cycle.
We have been experimenting with breaking that in recent years though. For most developments now, there's a socialised housing requirement, which puts poorer families into nicer surroundings. Of course the locals object to poor people bringing the area down, and given that the locals are more likely white middle class and the poorer families have a higher black/white mix that looks quite like racism too (although not institutional).
We still have idiots like that woman who said David Lammy can't be English because he's Afro-Caribbean and not Anglo-Saxon (that's German-German) like her, but they're shocking because they're so unusual and don't reflect us as a nation.
I don't disagree that the UK is classist, it most certainly is, but that doesn't exclude it being structurally and institutionally racist.
Plenty of evidence exists to show that it is, in fact, both.
One example of this is the Lammy report itself (many of the recommendations of which have still not been put in place), which looked at the Criminal Justice System. Only one area of the CJS has what the report described as a good degree of proportionality in outcomes based on race, which was the CPS, the CPS is also the only one of the areas of the CJS to have a workforce diversity that is in line with the country across all levels of its structure.
"One of the most notable features of the CPS, within the wider family of CJS institutions, is the diversity of its workforce (see Figure 2). The latest CPS workforce data shows that BAME staff account for 19% of those who declared their ethnicity.94 This makes the CPS one of the most diverse institutions within the CJS – it is, in fact, more diverse that the population as a whole (BAME people made up 14% of the general population, according to the 2011 census95). Significantly, this diversity runs throughout the organisational structure – for example 15% of Senior Prosecutors in the CPS are BAME.96 This contrasts with other parts of the CJS where BAME staff are much less likely to be found in senior positions within the organisation"
Its not exempt from criticism, but this does indicate that class and wealth alone are not the only factors at play.
"In most cases, defendants’ ethnicity does not affect the likelihood that they will be charged by the CPS. Other institutions in the CJS should look carefully at the factors that have driven this, from internal and external oversight, to a workforce that reflects the society it serves. There are some areas that the CPS should address. These include worrying disparities for the specific offences of rape and domestic abuse, and the role of the CPS (alongside other CJS institutions) in tackling gang crime effectively and proportionately."
https://assets.publishing.service.g...ata/file/643001/lammy-review-final-report.pdf
A similar pattern emerges when at trail by Jury vs. Magistrates in the UK. Juries, when they are representative of the community in question, on balance deliver proportionate verdicts regardless of race, Magistrates do not, not even close, with a massive bias against those who are not white in verdicts. Now the report does also point out that Magistrates are underrepresented in term's of working class backgrounds as well, that however supports both assentation's, rather that eliminate one or the other.
It's also important to understand that the two are interlinked, however one is easier to escape from than the other (which is impossible). I grew up in a sink estate, I was fortunate however to never fit into a stereotype (good for ASD for once) that saw me dress like my peers, and as such I never outwardly appeared as lower working class (or as it would be termed Chav). As a result I've only every been stopped by the police once in my life and wasn't searched, but rather asked two questions and allowed to go on my way, this is despite being clearly and obviously off my face. In short, my class has been easy to 'hide' from an early age and now I sit firmly in the middle class it simply isn't an issue. No assumptions about my background, past, etc. will follow me at all, and unless I volunteer them no one would know.
The same is not true for anyone non-white, who may or may not fall into two groups subject to barriers, black and poor or black and affluent, it doesn't actually matter what the former does to appear like the latter, a perceptive link exists in UK society that will still see them in the same way.
This is further borne out by the excellent paper that the Runnymead Trust produced with the LSE, showing the interlink between racial inequality and class structure, one important part of which is that mobility results at the lowest class levels are still higher (and out of proportion) for white's versus any other ethnic or racial group (despite better academic achievement in comparison).
To cite the author:
"As a result, each time the boundary of the nation was extended to more members of the working class, this was accompanied and legitimized by a racialized nationalism that excluded more recent arrivals. This dual process of democratization and racist exclusion was to be repeated throughout the twentieth century, with different migrant groups and their English-born children in the firing line each time."
This has happened with every 'wave' as the author lists:
"Critical to this process of class formation which went through race, not around it, was a social actor that I have termed the racialized outsider – who in different historical periods happened to be Irish Catholic, Jewish, Asian, African and Caribbean."
It's also why the two factors are so interlinked, class structures themselves are sub-divided, and it's in the interest of the class that wants control that this is maintained, and a new 'lower' tier is added, and that divisions between those are created when possible.
https://www.runnymedetrust.org/uploads/publications/pdfs/Race and Class Post-Brexit Perspectives report v5.pdf