Whether Rose is, or is not, an "accurate" working class character is fairly irrelevant (for my purposes here); the show isn't about working-class people; it's about a Timelord who travels through time and space in a magic box that bends reality.
"Defend the earth from alien invasion" or "teleport the crew of this spaceship to safety" are not working-class plotlines.
I haven't decided what I think of miniseries, but I suspect they're in the same mental category as movies: one-shot, intended as a temporary diversion, not a regular consideration. If "Boys from the Blackstuff" was planned as a continuing series, and canceled after a short run, it seems that'd be a good contender for lower/working-class TV series.
(Obviously, I need to acquire access to some British TV shows.)