It's neither American nor British, nor any other dialect of English. If you start out as a pop singer in a non-English-speaking country, it's the way you learn to pronounce song lyrics and possibly English words in general - because that's what plays on the radio. If you choose to use an actual dialect or accent instead, no matter whether it's your own or a different one, you avert this trope.
Typically, bands and singers who intend to emancipate themselves from their origins and local dialect (or accent) will often end up sounding like this, especially if from the British Isles or Europe and trying for a generic American accent. The bands of The British Invasion attempted to do this to Americanize themselves and, by being ridiculously successful and influential, made this kind of pronunciation popular and cemented it as a standard. As a consequence, the Americanization aspect may be completely absent today. Of course, American singers may employ this accent as well.
Another theory about the origin of this trope is that American vowel pronunciation is easier on the human vocal tract than the vowel systems of other dialects and therefore more suitable for singing.
As a rule of thumb, this type of English can be described as non-rhotic (no "postvocalic r") with American vowel pronunciation. Keep in mind that this is not an actual American accent but an artificial dialect that is normally only used in singing.
List of phonological features (starting with the most common conventions):
- The "a" in "dance", "after" etc. is pronounced as in General American (higher tongue position).
- The diphthong in "only", "road" etc. is pronounced as in General American ("o-u" rather than "e-o" or flat "o").
- The "r" after vowels before consonants is omitted as in BBC English (non-rhotic), which makes singing easier.
- The final consonant of a line's rhyme is not fully pronounced or completely omitted (as opposed to classical musical training).
- "New", "stupid" etc. are pronounced as in General American ("noo", not "nyoo", and "stoopid", not "styoopid").
- The "o" in "lot", "body" etc. is pronounced as in General American (lower tongue position and long, like "palm").
- The "ing" verb form is rendered "in".
- The vowel in "caught", "walk", etc. is either diphthongized to sound like the one in "mouth" or merged with "lot".
- The long E sound is changed from a high to a mid or even low vowel ("me" becomes "may"). Lower vowels are easier to sing (larger passage for air to flow through), especially at a high pitch or volume.