You know, I've been wondering if it's more a peculiarity of American television rather than television in general that working class characters don't have dramas about them -- I can think of half a dozen serious British shows that focus primarily on working class occupations and concerns off the top of my head that haven't been mentioned here: Jane Hall (dramedy about double-decker bus drivers); Sorted (mail carriers); Bad Girls (women's prison, mostly working class characters); No Angels (working class nurses); Clocking Off (factory workers); Skins, which has a broad swath of economic backgrounds in their characters; and Ghost Squad, which IS about the police, but deals mainly with how bobbies try to navigate council estates. Or something like Hotel Babylon would work, too, because though the guests are all very wealthy, most of the main characters (hotel staff) are definitely not.
And those are mostly contemporary or close to it, so I've not delved into the million and a half working class dramas from Thatcher-era Britain. In fact, my overall impression of British television is that their sitcoms tend to deal with "stupid rich people" and their more serious shows tend to address real drama, whether it's working class life or something else entirely.