Leetcode Count of Smaller Numbers After Self solution
The first advice is to remove the class. In python you should use classes when it makes sense to do so, but you don't need to.

Next, your bst_insert has redundant code. root.left = None and root.right = None are already done in your constructor, so you don't need to duplicate them.

The last piece of advice is to use reversed(nums) instead of num[::-1] as the former does not make a copy, which should be faster and use less memory. 

Other than that this looks pretty good.