I'm getting this problem too. I've downloaded the installer as of 28 November 2015 (version {ta_m-id%3A918-FOD-433%26token%3A_mch-autodesk.com-1384021993765-37031}).
I think that these problems started for me when I opened the software whilst an update was being pushed (on the 21st? Nov) and I got a message that the software was being updated and that I would have to work offline. I did the work that I needed to finish, but then every time I've started Fusion in the week since, I get the dialog that says that I need to update the software but when I clicked the button to carry out the update, the progress bar stayed fixed at 0% complete for hours. This evening, I'd had enough and figured that I'd nuke the software and reinstall. The uninstaller failed to work (it just opened and then instantly closed), and the downloader did the same. I then tried the instructions to carry out a clean uninstall of the software but they've not helped.
My machine is running OS X Yosemite 10.10.5 (14F1021). The Mac App store version seems to be working, but I need to maintain compatibility with Fusion running on Windows, so the update delays that the MAS adds for Apple's verification process mean that my copy of Fusion on the Mac often can't open anything that I edit on a Windows PC since changes to the file format (or whatever Autodesk change) makes files saved in the newer versions not backward compatible.
Here's the tail of the autodesk.webdeploy.streamer.log
2015-11-28 23:58:06,789 - MainThread - adsk.dls.streamer - DEBUG :: Logging initialized, with level 10, output file: /Users/timini/Library/Logs/autodesk.webdeploy.streamer.log
2015-11-28 23:58:06,789 - MainThread - adsk.dls.streamer.manifests - DEBUG :: Loading build manifest from /Volumes/Autodesk Client Downloader 1/Double Click to Install.app/Contents/MacOS/resources/buildinfo.json
2015-11-28 23:58:06,790 - MainThread - adsk.dls.streamer.manifests - DEBUG :: Type: <class '_io.TextIOWrapper'>, Members: ['_CHUNK_SIZE', '__class__', '__delattr__', '__dict__', '__dir__', '__doc__', '__enter__', '__eq__', '__exit__', '__format__', '__ge__', '__getattribute__', '__getstate__', '__gt__', '__hash__', '__init__', '__iter__', '__le__', '__lt__', '__ne__', '__new__', '__next__', '__reduce__', '__reduce_ex__', '__repr__', '__setattr__', '__sizeof__', '__str__', '__subclasshook__', '_checkClosed', '_checkReadable', '_checkSeekable', '_checkWritable', 'buffer', 'close', 'closed', 'detach', 'encoding', 'errors', 'fileno', 'flush', 'isatty', 'line_buffering', 'mode', 'name', 'newlines', 'read', 'readable', 'readline', 'readlines', 'seek', 'seekable', 'tell', 'truncate', 'writable', 'write', 'writelines']
2015-11-28 23:58:06,862 - MainThread - adsk.dls.streamer.configuration - DEBUG :: Loading default language: en-GB
Any ideas?
Tim