No. These are run by Transport for London (TfL) which is a government organisation.
The Overground is run on some type of agreement with TfL and Network Rail which is different to a normal rail franchise. TfL then awards a concession to a successful operator (Previously MTR and Arriva, now only Arriva). The system is branded by TfL as Overground. The operator provides staff and ticket machines. Network Rail deals with the track maintenance.
The Underground is a lot simpler. Around 10 years ago it used to have operators but TfL bought them out and now it is public owned. Underground is just Underground, there's no complicated structure.
Not sure about Crossrail but I think it will be similar to the Overground, if I remember correctly MTR has been chosen as an operator for it.