Nice work Oliver.
I eventually found the class data, although it only covers England and Wales. This is the NRS classification of social class grouped into ABC1 and C2DE, which is sometimes equated to middle and working class respectively - although, as
@iainbhx has mentioned here a few times, this is really really questionable nowadays due to many of the metrics being very dated and not taking into account how the nature of employment has changed. Nonetheless, it is worth mapping because--as with the ethnicity map I did--this tends to be a chosen metric for UK pollsters to break down their data by group, for better or for worse.
I deliberately chose the blue/yellow colour scheme to see if there was correlation
with the EU referendum - the answer is 'yes, but' as the broad pattern looks familiar, but the South is obviously far more middle-class than it was Remain, as is North Yorkshire and the rural bits of the Midlands. The curious concentration of ABC1s in Wimbledon and Richmond Park also doesn't correlate to the referendum result, which was strongly Remain there but not so much as it was in inner London. Obviously there are also plenty of other factors in the mix too. The closest correlation between ABC1/Remain and C2DE/Leave is seen in Wales, North East England, Cumbria and Cornwall.