The title itself is the thing i wanted to ask
It only changes is that you have now bought a copy of the music in the game which means you can listen to it anytime you want to on any computer with steam installed that is logged into your account, sadly i don't know how you access it or download it because i haven't bought the DLC.
While the soundtrack itself should appear in your Steam music library, you can find the files themselves in ...\Steam\steamapps\common\Tower Unite\Soundtrack in both .mp3 and .flac formats.
I bought the OST. It's great of course, you get to listen to your favorite music anytime but, I wonder they did not add music of Khromidro, Oasis and Prism in it.
The OST will have all of the TU soundtrack on it once it's finished, but it has a reputation of being slow to add newer songs to its listings.
these tracks are present on the bandcamp and google play music versions of the soundtrack. the steam version is updated less, as it requires going through a semi-lengthy process with valve.