There are not really any "very" diverse and/or predominantly African-American upper-middle class communities in the Nashville metro (at least not like you'll find in LA). However, there are plenty of moderately diverse upper-middle class neighborhoods that where your family will surely be accepted sans problems. Nashville simply doesn't have the high percentage of African-Americans like Atlanta or Memphis or the large numbers of a city like LA or Chicago. However, this remains a great place for families of all races and ethnicities.
There's an increasingly strong and visible number of African-American professionals here whose families live all throughout the metro. I work for a primarily African-American law firm so I see them all the time. If you're looking for a more urban, progressive neighborhood, you might find what you're looking for in places like East Nashville, Germantown or Hillsboro Village. If you prefer something more suburban, then Franklin, Hendersonville, or Brentwood are certainly worth considering. There are plenty of great neighborhoods in the Nashville area that'll likely work well for your family - just not really one in particular that's a majority minorities.
The only predominately African-American Nashville middle/upper-class neighborhood that I honestly know of is a tiny enclave usually referred to as in Enchanted Hills in northwest Nashville nestled between Bordeaux (predominately Black community but certainly not what you're looking for). I actually grew up in that area and I can honestly say that, unless you're dead set on living in a predominately African-American community, I wouldn't recommend it. For one thing, it's a small area and the nearby Bordeaux area certainly won't be your cup of tea. There are also very few nearby amenities (shopping, recreation, etc.). Most importantly, the zoned public schools in the area are amongst the worst in Davidson County. There are probably pockets of mostly upper-middle class African-American families scattered throughout the Nashville area in other places but I can't think of any right off.