There does seem to be some uncertainty about its set up. Interesting disclaimer on the website for it:
Disclaimer: The Queen’s Commonwealth Trust (QCT) is an independent organisation. Views expressed on the QCT website do not represent views of the Royal Family or the Royal Household. The QCT website offers a free and open forum where young people can share insights, expertise, advice and inspiration and come together to discuss issues that matter to them. The content and discussions shared by QCT are only intended to reflect the perspectives of those engaging with the platform.
Edit - its 2018 annual report starts with the following:
We launched The Queen’s Commonwealth Trust (QCT) on HM The Queen’s 92nd birthday, 21st April 2018. We wanted this Trust to be a gift to Her Majesty that recognised her lifelong belief in young people as a force for good in the world.
This belief was previously reflected in the work of The Queen’s Silver and Diamond Jubilee Trusts, where young people and their leadership in the community played a central part. The Queen’s Silver Jubilee Trust has now wound down and The Queen Elizabeth Diamond Jubilee Trust will end its work by January 2020. We wanted to create something new, established in perpetuity, that would speak to the new generation of young leaders – a place where they could come for advice, inspiration, connections, funding and support. I
That is from Lord Geidt (HM's former Private Secretary) and chairman of the QCT.