Time Machine stopped working after upgrading to Yosemite

I had the same problem. At the Apple troubleshooting site, it stated that the initial backup after an upgrade will take a long time. I hit "Back Up Now" before bed, and when I got up in the morning, it had completed the Back Up. After the initial Back Up, it works just fine now!

I had the same problem. I don't know whether turning FileVault on was related (this process was suggested during Yosemite installation). However, I retried backup when FileVault finished and it worked that time.

Remember that Yosemite installation brings around new 6 GB that will be backed up. So, the preparation of backing up will delay more than usual.