I have downloaded the profile for iPadOS on my iPad mini 5. But in Software-update iPadOS 15 does not appear. It say "iPadOS 14.7 iPadOS is up to date". (So it is a beta of 14.7, is that part of the problem?)
What must I do to get iPadOS 15 on my iPad?
I had a very similar experience when trying to get iPadOS 15 beta 2 installed from my Mac which was running Monterey beta 2. I just kept trying, expecting to get different results. And lo & behold - after 3 days - I did: it finally just worked. My only suggestion is to just keep on trying…
The rather comical account of my trials: https://developer.apple.com/forums/thread/683943 (Note that the answer to this thread includes the exact steps I repeated until I finally got success.)