I’m an American who thinks that American TV has gone to shit at least 10 years ago. Around that time I discovered the world of Korean and Japanese TV shows and movies(but heavily biased toward Korean, like 95%-5%). It’s a huge, refreshing change when you first start watching it, but after a while you start to notice that the Korean stuff has it’s own pattern. But I still really like it. The #1 reason to love Korean TV series is because they always end! None of that “canceled after airing one season, and ending on a massive cliff hanger shit!!!” that happens to the majority of American shows. Even if the show is getting low ratings, they soldier on and finish what they started. Oh yeah; and every fictional series is always one season! It might be as few as 10 eps or as many as 54 eps, but they air 1 or 2 eps a week until they are done and that’s it! Another great reason is that 99% of the time the actresses are just so beautiful! ;-p I’m a Gen Xer so I think women with tattoos and/or non-ear piercings are trashy as hell. I can only think of one time I saw a woman with a tattoo; it was a Korean movie and she got one and her friend got a matching one.
Okay, enough about that, I’ll get to responding to the main point of your post. I’ve got a lot of great suggestions, IMHO of course. 
I’ll get the two British ones out of the way first:
Jekyll - It’s actually a 6 episode mini-series instead of traditional TV series. It’s a modern take on the Jekyll & Hyde story and it’s damn good!
Coupling - By sheer coincidence, this show was created by the same man who did Jekyll; one of the stars of Coupling even stars in Jekyll. Prior to watching this series I thought SOAP was the funniest TV series I had ever seen. But Coupling managed to blow away even SOAP; no matter how many times I watch it I laugh. He was paid to attempt an American version of the show which was truly awful and should be avoided at any cost. There are 4 seasons, but remember that the UK considers 6-9 eps to be a full season, so there aren’t really very many eps. This series by far is the one that I would recommend most to anyone reading this if they will only watch one series.
Okay, now all the American shows…
Dead Like Me - Two seasons and a movie. The series is fantastic but the movie sucked, mostly because two of the characters were replaced. But it did give some closure so it’s worth the watch.
Wonderfalls - From the same guy that made Dead Like Me. Only one season before it was cancelled. But luckily instead of ending on a cliffhanger, the last ep can actually be looked upon as a satisfying series finale, IMHO.
Z Nation - Who knew the SyFy channel was capable of making something that didn’t suck like a dozen sharknado movies? This was done during the Walking Dead time, but with a good amount of humor mixed in with the drama and action. Lots of great characters and interesting stories. The bad part? Aired for 5 seasons and didn’t get a proper ending.
Falling Skies - Alien invasion series that was relatively more realistic than other alien invasion series or movies. Pretty damn good SFX for a TV series. Used long story arcs instead of self-contained eps, which helped the show feel more immersive. Ran for 5 seasons and got a proper ending! The bad part? The ending was boring and formulaic; it almost ruined the 5th season. Still definitely worth the watch though. A nice bonus is that it stars 3 really beautiful actresses, which of course never hurts a show.
Invasion - Was shitcanned after one season on a huge cliffhanger, so I can’t tell you if it was an alien invasion show or not. But it has good characters and an interesting story. Am still really pissed it got cancelled. It takes place in Homestead, Florida, but obvious Los Angeles sets were obvious, especially the sets that were supposed to be the Everglades.
American Gothic - Sheriff of small town appears to be Satan? Or maybe just a demon since he doesn’t seem very powerful a lot of the time… He targets a young boy who is somehow a threat to him. The boy has a dead sister whose ghost he can see and tries to help him. As usual, ran for one season and ended on a cliffhanger. And CBS even aired some of the eps in the wrong order which made it confusing to watch! The DVD release corrects that at least. Seemed like it could have been really good but didn’t get to run long enough to find out. Not a must-see show, but recommended if you like paranormal shows.
Space: Above And Beyond - Space sci-fi show that was on in the beginning years of the Fox network. Humans have traveled to other planets but haven’t found any aliens yet. That changes when a human colony is wiped out. So we go to war against an enemy that no one has seen in person and lived to tell about it. The story is about a squadron of fighter pilots aboard a space aircraft carrier. Good SFX at the time and the show was definitely interesting. And you guessed it; it ran for one season and was shitcanned…
Jericho - Conspiracy show about some secret govt group that nukes like 20 major American cities so they can take over the country. The show takes place in the small town of Jericho, Nebraska, and follows how they handle it. Cancelled after one season naturally, but a letter writing campaign got it brought back for a half second season. And no, of course it didn’t get an ending.
The New Outer Limits - Everyone knows about the old black & white TV series The Twilight Zone and The Outer Limits, but in the 90s there was 7 seasons of Outer Limits made. There are LOTS of REALLY great episodes in those 7 seasons; I highly recommend this series if this kind of show is your cup of tea.
The last one I can think of is one I mentioned with Coupling; SOAP. 70s TV series that was a parody of daytime soap operas. 4 seasons of funny storylines. And as you already have guessed by now, it ended on a huge cliffhanger. It wasn’t cancelled because it had low ratings, it was because some bible-thumpers got offended by it and threatened to boycott any advertisers. It worked. So the first case of cancel culture is older than most people know…
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not as great but a good watch in the same line - only 6 epis: London Irish.