Thanks again, @mike2 
So, If I understood correctly, based on all your previous messages:
1 - The first logical storage drive - /dev/sda - is a "Samsung model: SSD 860 EVO 500GB size: 465.76 GiB", with type "dos" - that is, it has a MBR (Master Boot Record) partition table - and has only one partition - /dev/sda1, formatted as ext4, and has a label of "System backup". You use that SSD for saving your backups, right? Is that an internal SSD -OR- is it an external SSD (connected via USB)?
2 - The second logical storage drive - /dev/nvme0n1 - is a "Samsung model: SSD 980 PRO 1TB size: 931.51 GiB", with type "gpt" - that is, it has a GPT (GUID Partition Table) - that has 2 partitions:
-
/dev/nvme0n1p1formatted as "vfat" where the ESP (EFI System Partition) resides and that is mounted on/boot/efi -
/dev/nvme0n1p2formatted as "ext4" that is mounted on/that has your Ubuntu MATE 24.04 LTS ("Noble Numbat") installation.
Did I understand correctly your setup?
If I did understand correctly your setup, I would do what I describe below. There are some risks involved, so please FIRST make sure that you have some kind of USB flash drive with Ubuntu MATE 22.04 or Ubuntu 24.04 that you can boot from (in case you stop being able to boot from your hard drive) AND, preferably, also guarantee that you have a good BACKUP of your data in some kind of external drive!
1 - You've mentioned that you have your UEFI settings changed to Legacy. But, according to your setup, it seems to me that is not necessary, given that your Ubuntu MATE 24.04 drive has a GPT (GUID Partition Table) and an ESP (EFI System Partition). Is there any particular reason why you changed it to Legacy?
2 - I would do the same changes to the /etc/default/grub file in your Ubuntu MATE 24.04 installation that I had suggested in my previous message for your Ubuntu MATE 22.04 installation and, based on your setup, I would also UNCOMMENT (that is, REMOVE the initial # sign to activate it) from the line #GRUB_DISABLE_OS_PROBER=false. So, change the following line:
#GRUB_DISABLE_OS_PROBER=false
... so it becomes the following line:
GRUB_DISABLE_OS_PROBER=false
So, please edit that file with administrator privileges, by running the following command (that will open the "Pluma" text editor as superuser):
sudo pluma /etc/default/grub
In that /etc/default/grub file, change the following lines
GRUB_TIMEOUT_STYLE=hidden
GRUB_TIMEOUT=0
GRUB_CMDLINE_LINUX_DEFAULT="quiet splash"
#GRUB_DISABLE_OS_PROBER=false
... so they become the following ones:
GRUB_TIMEOUT_STYLE=menu
GRUB_TIMEOUT=10
GRUB_CMDLINE_LINUX_DEFAULT=""
GRUB_DISABLE_OS_PROBER=false
After doing that, save and close the file, and then run the following command:
sudo update-grub
Please, copy the output of the execution of that sudo update-grub command to some text file, so you can also include that output in another reply from you in this same discussion topic.
Now, please reboot the computer. During the reboot, go to your UEFI Settings and change it from Legacy to UEFI, save your changes and exit the UEFI. After that, see if the GNU GRUB boot loader menu appears and what are the options that appear in that menu. If possible, take a picture of the screen at that time (for instance, using the camera of your smartphone) and please reply back here with the result. And again: Good luck!