I have solved my problem.  While working on a different issue with the laptop, I came across a process to reset the BIOS by unplugging the main laptop and CMOS batteries, pressing the Power button for 30 seconds, reconnecting the batteries, and starting the laptop.  After that, I was able to update the BIOS from Windows 10 using the BIOS package I downloaded from Dell.  I assume that the BIOS update will work from BIOS as well, but I did not try it because I did not need to.
I found 1 old post with this same issue; there was no resolution so I am creating a post. I have a used Latitude 5480 that I just purchased.  I installed an M.2 SSD, loaded a clean install of Windows 10 Pro and completed all of the updates (no missing devices in Device Manager).  The current BIOS version is 1.19; I downloaded 1.29 from the Dell website and attempted to update it.  I've tried to update via running the BIOS exe within Windows, using the Dell Command Update application within Windows and via USB at boot time.  There's no failure warning or prompts the laptop just reboots normally into Windows 10 without going through the usual BIOS install routine at reboot.  The old post recommended checking to make sure BitLocker is not running; it is not enabled. I also tried to install previous versions of the BIOS (1.14, 1.15) to see if it was an issue with 1.29.  Unfortunately, I get the same result.  It starts the installation process and restarts the computer but the BIOS remains 1.19.  I don't see any settings in the BIOS that prohibit it from being flashed, but maybe I am missing it.  Any ideas would be welcome.  Thanks!
Latitude 5480 will not update BIOS