So in both cases Ubuntu MATE is the only OS installed on the boot disk - correct?
If so possibly GRUB menu is showing and waiting for input [10 seconds]
Please paste the output from inxi -ACDMNSG
one of your machines [we'll work on 1 at a time].
In order to use this command enter a MATE terminal by either:
- Accessing menu Applications > System Tools > MATE Terminal
or
- pressing Ctrl+Alt+t keys simultaneously
use the code tag </> [see below] to mark the output