I have tried doing searches for this and can't seem to track down the exact numbers. The sources will list the top selling genres but that doesn't include children's books.
Since you don't specify which market you are interested in, I assume you want to know about Germany ;-)
In Germany, the book market is divided into the following merchandise groups:
The 15.8% market share of books for children and adolescents is in turn subdivided into the following merchandise groups:
Please note that "Learning" and "Non-Fiction" are both comprised of books for children as well as adolescents.
Also, there is no "Middle Grade" segment in Germany as there is in the US. Middle Grade books are subsumed under the children's and adolescent fiction categories.
There are no numbers for self-published books.
(Statistic: Börsenverein des Deutschen Buchhandels; Chart: self-made)
Unfortunately, I did not find any hard statistics in my searching. I assume someone tracks these sorts of statistics but may not make them readily available to the public. You might want to contact the Society of Children's Book Writers and Illustrators (SCBWI) or Publisher's weekly.
SCBWI offers resources on self-publishing for children's lit authors. I did not see the statistics you want in their publications but someone there might have that information or know where to get it, since they are dedicated to helping authors and illustrators get published.
I'm not as familiar with Publisher's weekly but it is an organization dedicated to all things publishing, including self-publishing. They are a likely source for the information as well.
What I'm seeing around seems a few years old with a quick google search, but this Statista website may have a report to answer your question or something to get you started at least.
Hope this helps if you're still looking for an answer.