The opening of the UK labour market to any EU citizen (indeed the opening of settlement rights to any EU citizen) has expanded greatly the presence of people working in unskilled categories D, E or not working at all. It is these categories that slipped most in the Labour vote, though C2s disappeared too.
Much hand-wringing has gone on in Labour discussions of why-we-lost about the turning away of the WWC vote. It is worth noting that voting rights in a UK general election are not conferred by EU citizenship and that there are reservations by many EU nation states on their citizens holding the citizenships of other nation states without losing their own (though not on their EU citizenship statuses).
Lots of potentially Labour voters cannot vote in a general election in the UK in spite of Labour's brechtian efforts in dissolving the people. Labour needs to sort that and get all those C2,D,Es on the register asap. Unfortunately (for Labour) doing so would open voter registration in the UK to further unwanted scrutiny and might even lead to a European-wide standard for electoral registration that would take more from Labour than it gives.
Identity cards would make an unwelcome policy return as well.