Firstly, I don't really know what others have been saying about the accents. They are all pretty solid - the character who they have been complaining about is an intentionally offensive redneck stereotype. Someone else here has complained of the show's low budget. What do you expect? It's a small relatively unknown program in its first series. That being said, initially lower- budget props and sets are replaced in later episodes with filming locations of increased diversity and much higher quality set-pieces. Noticeable or bad CGI is minimal, if there at all.
Anyway... an overview...
Throughout the show, several characters around the ending world have their stories intertwined in what, to be honest, seemed like a boring cliché to me. Then I watched it. I was really, really, really wrong. I mean, I guess the show's subtle humour wouldn't appeal to everyone (the delivery reminds me of Parks and Rec without any fourth wall- breaking), but it's dispersed very naturally in each episode, along with many striking scenes that can be both tantalizing and ridiculous. It is a genuinely new take on the apocalypse genre: something fresh, innovative, and really just great fun to watch.
The main area in which the show shines is in character writing and development. Well, that, and the spiraling web of interconnected story lines that to follow is engaging, moving, and still quite believable. Increasing with each episode is the depth and realism of every character's motivations: their personality; their morality and their sanity. 'You, Me and the Apocalypse' undoubtedly has some of the most diverse and brilliant casting in recent comedy dramas (in terms of fame, race, age and gender).
There's not really much else I can think of that doesn't delve into spoiler territory. SO WATCH IT.