After a month of beta testing, the watchOS 5.1 update went live for Apple Watch alongside the release of iOS 12.1 for supported iOS devices. For most Apple Watches users, the watchOS 5.1 update installs just fine. But for the Series 4 Watch users, the update gets stuck on the Apple Logo after installation.

Many users have reported the issue on the Apple Community forums. One user 
gtdandy who updated his Apple Watch Series 4 to watchOS 5.1 says:

Been using and enjoying the latest Apple Watch for the last few weeks (44mm, with cellular). Started update via iPhone app, on a phone which had already been updated.
Checked watch update progress about 90 minutes later and Apple logo was visible, but no update ring was displayed. Waited longer to see if the watch would complete its update, but no luck. The watch was unresponsive to screen or single button presses, so I tried a hard reset. Watch resets, but is stuck displaying apple logo.
Will likely visit an Apple store tomorrow.

Another user zackfromaurora says he waited for 5 hours to get his apple Watch series 4 to boot, but it is just permanently stuck at the Apple Logo.

If you own an Apple Watch series 4, we advise you to not update to the watchOS 5.1 until Apple fixes the problem. For the folks, who have already updated and are stuck at the Apple Logo, you need to visit an Apple Store to get your watch fixed.