How I wish i'd researched about Big Sur first before finally updating yesterday. I thought everything is okay by now since this has been available last year. Same issue here, I cannot see my external drive, instead I can only see an external drive named "Update". Because of this, and other issues on email, photos, missing folders and very slow speed, I reverted back to Catalina via Startup Disk (Catalina on Macintosh HD vs Big Sur on "Update").
Like the others here, in the Disk Utility, I was able to see my originally named external HD "Touro MacBook BU". I used First Aid for the external HD but I still couldn't see it on Finder.
Here is the workaround I did to access my external HD files:
> in Disk Utility, right click on your external drive ("update" will be there too) and select Show in Finder.
> a Finder window will open your external drive and show folders App, boot, Lib, Sys, Users.
> unhide hidden folders by pressing "command shift dot"
> you will find a hidden folder named Volume, you should be able to see your Data folder in there containing your files.
This is such a hassle because you can only access your External HD via the Disk Utility :( it is the only place where I was able to see it. I hope "Big Sur" acknowledges this issue and finally provides a fix.