Using the solution mentioned here https://us.forums.blizzard.com/en/wow/t/battlenet-did-not-auto-update/1794027 by Vortimer on the WoW Forums has resolved this for me, so far, and I hope it does the trick for you all as well. 
Oh, right, I can’t post direct links - if you’re not comfortable copy/pasting it, just search on the US WoW Forums for something like “Bnet version” (without the quotes) and it should be near the top of the search results.
Basically, just manually update the Bnet app by downloading the installer, running it, accepting the UAC prompt, and it should update your Bnet for you no problem.