Even when doing minor updates, not just major version upgrades, it is important to make sure there are no significant errors when updating grub. If you do get errors, it is sometimes possible to fix them before rebooting.
If you know there is an issue, instead of rebooting at the end of the upgrade, use the command prompt to rerun the grub installer. Even if there are no errors, it is generally safe to do this. The commands in ubuntu are:
The first command updates the grub menu and will find new kernels and remove ones that are no longer available. The second command will reinstall the EFI bootloader. If you are still using a legacy bootloader, you need to specify the drive to install the bootloader on with this command. (For legacy, you can run it multiple times, one for each drive if you which boot off of multiple drives for redundancy, and your bios supports it.)
When you rerun these, you may get errors, and those errors may guide you to what you have to do to resolve the situation. For example, I recently had to resize my EFI partition because the grub bootloader has grown. (Luckily, it was adjacent to a swap partition, so this was not too difficult.)
Note that the above commands do not fix problems if individual kernel installations get errors. For example, if you have a separate /boot partition, and it fills up, you will need to reinstall any kernels that got updated when it was full. You can fix this either by resizing the partition, or by deleting older kernels that are not being used. (Easiest is with apt autoremove or apt remove, but you can remove pieces by hand with some care in an emergency.) Once space is available, you can use dpkg-reconfigure for each affected kernel package to trigger a rebuild of the initrd and copying of kernel pieces. Also, sometimes apt remembers what failed, and rerunning apt upgrade may give directions on what to do next to rerun the failed parts.
If The Ubuntu 18 to Ubuntu 20 upgrade triggered an upgrade of grub legacy to grub2, there are guides online about how to fix issues with this. The transition manager allows you to leave both bootloaders installed, so you can test the new one but fall back to the old one if it fails. Once you've succeeded booting with the new one, it is safe to finish the steps in the guide to remove the old one.
Note that if you already made the mistake of rebooting and the boot fails, and none of the options left in the grub menu boot successfully, worst case it is possible to use the ubuntu 20 install media to boot in rescue mode and attempt repairs. The rescue mode has an automated repair in it that sometimes works, but you may need to drop to command line, at which point it would be possible to free up space (or repartition -- but this is tricky) and try repair again, or chroot into the installed operating system and retry the steps as outlined above to attempt to manually effect a repair.
Note that repartitioning disks for an installed OS or hand removing kernel pieces in /boot are expert level operations and require extreme care and full understanding of all the consequences.