Even after controlling for work region, and for other variables, a sizeable gap remains, at least according to this government report.
https://assets.publishing.service.gov.uk/government/uploads/...
4.3 Explaining the Class Pay Gap
In order to disentangle potential sources of the class pay gap we next use regression analysis to see how the gap is affected when we adjust for five sets of factors often identified as sources of income inequality. Figure 3, begins by showing (in the first bar) the raw average earnings gap reported in Figure 2, while the second bar shows the average earnings gap in pounds per week when we control for gender, ethnicity, age and hours worked. In the next bar, additional controls for educational attainment are included; then human capital, work context (work region, industry, firm size, public vs private sector) and finally the individual occupation the individual works in.
[..] even when we control for all of these variables, Figure 3 shows that the class pay gap remains substantial and statistically significant (see Appendix Table A10 for standard errors). Thus even when those from working-class backgrounds are similar to those from advantaged backgrounds in every way we can measure, they still face a 7% or £2242 a year pay gap in Britain’s professional and managerial occupations.