We've watched a couple of British tv series over the past few months that are excellent and that I highly recommend. We weren't looking for British shows when we found these. We were just looking for quality programming and happened to stumble across these. Had never heard of them before I found them just browsing Amazon Prime Video.
Unforgotten - A 5-season British British crime drama series which follows a team of London detectives led by DCI Cassie Stuart and DCI Jessie James (Series 5) and DI Sunny Khan (Sanjeev Bhaskar) as they solve cold cases of disappearance and murder. I found out today that there will be a season 6. IMDB rates it 8.4.
Endeavor - A 9-season British detective drama series about a young Endeavour Morse who began his career as a detective constable, and later became a detective sergeant, with the Oxford City Police. IMDB rates it 8.6 (We are just now starting season 5 and we LOVE this show)
Annika - A 2-season Scottish crime drama series starring Nicola Walker who was DCI Cassie Stuart in Unforgotten. This series doesn't appear that it will be as good as the other two but we will probably watch it anyway because we like the actress. We probably won't start it until we finish Endeavor. IMDB rates it 7.1.
Other shows from across the pond that we've watched: Luther, Gangs of London, Peaky Blinders, The Night Agent. They were ALL really, really good.