Most larger mainstream news with political objectives can also be defined as fake news with a major political bias due to the profits they need to create to stay afloat. Business comes before accuracy in reporting. WSJ and BBC are probably great & accurate for stories about plants and polar bears, but anything of any major significance will be extremely biased in the favor of profit.
Never mind that BBC is funded by royal charter rather than by customers, so they get paid whether people read them or not.
You're confusing accuracy with bias. It is possible to give an accurate account of a story despite having a bias, so long as you're still stating the relevant facts. The bias comes from the interpretation and whether they state the facts in a positive or negative light.
As for biases, WSJ is known to be center-right, NYT is center-left, The Economist is biased towards classical liberalism, etc. When you read the stories you can take the bias into account pretty easily.
https://en.wikipedia.org/wiki/Propaganda_model