The general rule of Ayatollah Sayed Sistani is If the music that is suitable for entertainment and amusement gatherings comes out of it then it is not permissible.
These children’s rhymes do not seem to be the music that are used for night clubs for example. However the child listening to these and getting involved in these things may eventually lead them to want to listen to the other types of music when they get older. It is very difficult to navigate these things when our children go to school in the West and may Allah help us all. We should strive to teach them as much as we can at home without overwhelming the children of course. We can tell them that there is no benefit in these types of entertainment and the real benefit is in other things which lead us closer to Allah.