To clarify some questions here.
London Overground is operated by their concessionary partner Arriva Rail London under a contract that runs until May 2024 with the option to extend until May 2026.
Arriva’s responsibility is to manage the trains, stations on behalf of TfL who stipulate what services should be operated, down to the branding. It’s TfL’s responsibility to acquire trains and pay Arriva to run the service. There are similar concessions for TfL Rail and the DLR.
TfL’s plans for the Southern side include reverting back to six trains per hour into London Bridge off-peak (currently 4). I believe the extra two trains will go to Sutton southbound as pre 2010. while an extra two trains per hour would run between Dalston Junction and West Croydon and Crystal Palace. These combined services will provide a real tube style turn up and go service. They would also acquire new trains to replace the over 30 year old Class 455 trains which are used on Coulsdon Town services.
The extra trains into West Croydon won’t happen until the extensive resignalling and track works between Selhurst/Norwood Junction and East Croydon are finished.
The replacement bus services are contracted to Southern, rather than London Overground. As Southern is part of GTR, which is part of the Go-Ahead Group, they largely use buses from parent company Go-Ahead London.