The time to uninstall an older version of the application is BEFORE you install the new one.
Running an uninstaller for an older version when a new one is already installed carries a very high risk of damaging the new installation, since there are auxiliary files that are labeled the same or similarly and in the same or similar locations.
If you really want to remove the older version, your best bet now is to uninstall CC first, then uninstall CS6, run the Creative Suite Cleaner script for good measure, then re-install the CC version.
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Lightroom has never been part of the fake, artificial "suite."
Inconsistency between or among applications in the artificial "suites" should come as no surprise.
The "suite" concept is a fabrication of Adobe marketing and bean-counting types. The engineering teams are totally independent of each other, they are not only in different buildings but in different cities and states of the American Union, even in different countries.
The fact that they have little if any communication among them is highlighted by requests occasionally made in these forums by top Adobe engineers to let the other teams know when there are problems in one application that impact our workflow in another one.
There's a completely separate, dedicated Adobe Lightroom forum at:
Message was edited by: station_two
Incidentally, if you're running that tight on space, freeing up the relatively paltry amount occupied by CS6 (surely under 2 GB) won't make that much difference anyway.
What is too often underestimated by too many users is the amount of available disk space necessary for Photoshop's scratch file.
My personal rule of thumb is to figure on up to 50 to 100 times or more the size of your largest file multiplied by the number of files you have open.
Also, if you only have one drive, Photoshop's scratch file must share the available space with the swap files of the OS.
I'd like to keep a bare minimum of 250 GB to 300 GB available just as Photoshop's primary scratch disk on a physically separate, dedicated internal hard drive. I mention this just to bring into perspective how insignificant the drive space you'd gain by uninstalling CS6 is. But if you have no choice, then uninstall CC and CS6 as described above, run the Creative Suite Cleaner script, then re-install CC.
Please note the edits to my previous posts, which are not reflected or broadcast to those accessing forum messages by email.