Not sure if any experience like this has been posted, I found it difficult to find this information when I needed it, so I’m posting here.
I have a My Cloud Mirror with 2 x 3TB drives backing up a Mac via Time Machine. The hard drive on my computer died so I replaced it and reformatted (that is another story, if you have a fusion drive then you have to follow these instructions to recreate the fusion drive bw the internal SSD and the new platter-based drive you install and make sure you do not have an new OS X installation on it before you try: How to fix a split Fusion Drive - Apple Support (CA)).
When I tried to restore my Mac from my TM back up, TM could not read the folder to use it to restore over the network. I was worried the sparse bundle was corrupted based on some of the error messages I received but that did not turn out to be the case.
I cloned the Time Machine back up folder to another drive (in a UBS enclosure) using Carbon Copy Cloner. I’m sure there are other ways you can do this but I’ve used CCC before and trusted it. You have to format that new drive in HFS+ and I chose to make that format case sensitive based on a warning that I received when starting the clone. I ignored the other warning I received that the My Cloud Drive was a different (i.e. non HFS+) format before cloning (this, I believe is the reason that the TM back up was not recognized as a restore volume). To my knowledge, you cannot have a NAS in HFS+ format so this problem might affect other NAS devices and TM.
Once I had this clone on an HFS+ drive I could connect to my Mac by USB, I was then able to use the back up to restore the Mac entirely.
Hope this helps if you ever find yourself in my position.
Edit - this solution obviously only works if you have another computer to connect to your NAS to clone the TM backup folder. Based on what I have read, it may also be possible to do a fresh OS X install and then select the drive over your network using Migration Assistant to restore your files.
