Yes we can, sometimes! I think of Adele. Some British artists may let their accent through in some songs more than others. I think it partly has to do with how we feel music and how we feel most natural singing.
A lot of what has to do with producing 'good' sounds is forming vowels in a certain way. This will also have a huge impact on whether or not someone's true accent comes through. Vowels are ideally open and lengthened, so that the sound that comes through doesn't seem forced. You also have to concentrate on where you project sound - choral singers lift the soft palette in their mouths so that the sound resonates seemingly up through the top of the head, whereas pop singers using their chest voice want to push the sound forward. This is also an important factor in determining the sound of someone's voice - you don't want to sound nasal!
Hope the technical stuff thrown in at the end wasn't too confusing. But singers really do need to hone and craft their voices. Voice is an instrument.