Let me say upfront that had I tested the upgrade against clones of the productions VMs, I would have found these problems ahead of time (note to self for future upgrades!).
I followed the steps published in Its Time: vSphere 5 Upgrade so the first component to get upgraded was the vCenter instance at the recovery site. This completed successfully - no errors - so far so good.
Next up, vCenter upgrade at the protected site. The database upgrade step failed(!). The installer gave a fairly generic error: "database upgrade failed. See the log at...". The log only mentioned: "failed to execute dbuhelper.exe: an error occurred".
A quick Google search finds the log error posted in a VMware Communities post (TGfVC! - thank goodness for VMware Communities!). The resolution: set the log file growth to unrestricted. Sure enough, my log file was set to restricted so I changed it, re-ran the install and viola, success!
That evening, I reconfigured the vCenter Linked Mode feature but they never seemed to reconnect. By time I came in to work the next morning, it had started working. I must not have been patient enough(?), not sure but it has worked fine since.
Finally (for this step), vUpdate Manager upgrade. This went smoothly just as it did in the lab. Now that its working, I can start upgrading the ESXi hosts.