There doesn't appear to be a solution so far and Apple has now pulled the update. The problem has affected owners of several different Watch models, including the new Series 4, so if you managed to download the new release, it's advisable not to install it. If you really need your Watch and the update bricked it, you might have to contact Apple support to figure it out. Engadget has reached out to Apple to see if they have any comment or, better still, a fix.
Update 10/31/2018 4:22 PM: Apple has acknowledged the issue and given the following statement.
Due to a small number of Apple Watch customers experiencing an issue while installing watchOS 5.1 today, we've pulled back the software update as a precaution. Any customers impacted should contact AppleCare, but no action is required if the update installed successfully. We are working on a fix for an upcoming software update.