Until the advent of BBC America, most British shows were shown on PBS (Pubic Television System) and were/are considered a bit high brow.
Masterpiece Theater featured, for example, Upstairs Downstairs, The First Churchills, Poldark, Brideshead Revisited, Love in a Cold Climate (the one with Judi Dench). Downton Abbey is now Masterpiece's most popular series ever.
Later we began to get the "Britcoms," Fawlty Towers, Blackadder, Are You being Served, 'Allo 'Allo, You Rang, M'lord, Keeping Up Appearances, As Time Goes By. No soaps.
But PBS was very much for (and still is) a niche (pronounced "nitch" btw) audience, so many Americans would not have watched them.
Now we have much more with BBC America. Luther, Broadchurch, etc.
We also had and still have quite a lot of British mysteries: Morse, Inspector Lewis, Inspector Banks, Miss Marple, Foyle's War (my current favorite), Rebus, etc.