In my opinion, the television news shows may be too fast paced with too many verbal shortcuts to be good sources to listen to. To start with, you may want to try audio books where what is being spoken comes from a written text. I think that it isn't the accent that is problematic as much as the content and pace of the speaking. There are some British movies that I, as a native American-English speaker, have trouble understanding without subtitles while others are easier to understand than some American movies.
I think that dramatized audio is a good choice for listening skills because it is intended to carry all of the meaning through speech and the pacing is likely to be more deliberate. If you purchase an unabridged audio book along with the written book, you would have a transcript to look at if some particular passage was difficult to understand. The quality of an audio book can vary widely based on who is reading the book and how the recording was made, so I would look at reviews before investing too much money.
There are a number of sites that offer free audio dramas - I don't have one particular site I can recommend, but a search for "free audio drama" returns a long list of possibilities. If you like science fiction, The Internet Archive has episodes from "Seeing Ear Theater" that I enjoy, although not all of the recordings are great quality.