British TV series are shorter than US ones, usually 6 or maybe 12 episodes.
I'm the Brit in this relationship but DW likes a lot of British shows. Currently on - Strictly Come Dancing, Downton Abbey, Whitechapel
Recent crime ones include...
The Guilty
Broadchurch
Also
Spooks (called MI:5 in the US) but that has finished now.
Comedies -
We watch a lot of the satirical news shows - Have I Got News For You, Mock The Week etc
The Thick OF It, satire on politics, written by Armando Iannucci, who has also written Veep in the US, which is basically an American version of The Thick of It minus the swearing!
Just started watching The Wrong Mans, a dark comedy.
For older comedies check out the channels Gold or Dave, classic Fawlty Towers, Dads Army, Only Fools And Horses, Red Dwarf (sci-fi comedy), Little Britain.