Was that definitely Microsoft Support? As it's not too common an issue? Anyway, what make is the PC? If HP for example, their update Assistant can upgrade the BIOS, it's just a firmware update. Or Dell have their Dell Update software. And as above the Power needs to definitely NOT go off during a BIOS update, as that would be bricked motherboard!
Once done, then you can download a Windows update assistant for Windows 10, https://www.microsoft.com/en-gb/software-download/windows10
run it once downloaded and tell it you're upgrading rather than creating a bootable pen drive.It usually works ok, and I'm please you have everything backed up of importance!