Update: Dell support said that after all the troubleshooting I have done it is clear that there is a fault with the motherboard. So I replaced the motherboard and now it behaves as expected. I managed to update the BIOS from Dell Command Update in windows. After the reboot the BIOS was installed. All good. 
I am trying to update the bios on my Precision 5820 Tower, from 2.8.0 to 2.12.1. It just wont start installing. If I try from Dell Command Update, it reboots as if it is going to install the BIOS update, but after the reboot there is no sign of any BIOS update, and it simply reboots like normal, having installed nothing.  If I download the bios update file 5820T_2.12.1.exe from the dell website and run it, it does exactly the same. It reboots as if it is going to install the BIOS update, but after the reboot there is no sign of any BIOS update, and it simply reboots like normal, having installed nothing. I have also tried copying in the 5820T_2.12.1.exe to a FAT32 USB stick. Rebooting and selecting firmware update from the F12 prompt. Then I can see the BIOS update tool. I select the 5820T_2.12.1.exe from the USB and click through the warnings. Then the same thing. It reboots as if it is going to install the BIOS update, but after the reboot there is no sign of any BIOS update, and it simply reboots like normal, having installed nothing. I have also tried making a Dell bootable USB and tried to run 5820T_2.12.1.exe from DOS, but I get the error "This program cannot be run in DOS mode". I can confirm that BitLocker is off for all drives. I can also confirm that the service tag is listed correctly in the BIOS.  I can also confirm there is no other strange behavior with the machine.  I have also tried to install an older version of BIOS 5820T_2.10.0.exe, but I get exactly the same behavior.  I have also tried unplugging everything except a keyboard, a mouse, 1 screen and a USB stick. This includes removing all other USB devices, network cables, and all hard drives.  I have also tried all these settings in the BIOS: - UEFI boot path security - disabled - Absolute - disabled - Intel virtualization technology - off - Secure boot - off - UEFI - set to legacy None of the above, and none of the above combinations have any effect. Always the PC just reboots as normal.  Running Windows 10 Pro 21H2, fully up to date. 64BG RAM, Xeon W-2145. Some extra hard drives (but i have tried with them all out). The rest of the machine is as it was delivered.  I would much appreciate to hear if anyone else has had this issue and found a solution that is not listed above? Thanks!
Precision 5820 Tower - BIOS update does not start