Was able to do the updated for Lightroom CC this morning after I figured out you have to sign out of Adobe Creative Cloud and sign back in before the updates show up. After that all updates worked just fine.
Makes you wonder what the point of the Creative Cloud App is, from the customer perspective, if you have to keep logging in and out of it for it to do its job. Same happened on the release of LR CC. They updated the App twice (with the same version) a couple of days ago, too. All seems a bit chaotic.
Hope that the performance of 6.1 is better than 6 and that there are no avoidable glitches such as those that had to be fixed on 6.0.1.