Change the default editor when sudo visudo

The problem is not that it does not apply to nano, it's that it does not apply to the shell:

Just set the VISUAL environment variable:

export VISUAL=vim

Add this too ~/.bashrc to make it permanent.

As you seem to use vim in general, set both VISUAL and EDITOR:

export VISUAL="vim"
export EDITOR="$VISUAL"

or more POSIX-correct

VISUAL="vim" ; export VISUAL
EDITOR="$VISUAL" ; export EDITOR

I assume nano was the value of one or both variables.

To make use of the editor in visudo actually, we need to handle that sudo does not keep the environment variables by normally. The option -E changes that.

sudo -E visudo

Without the -E here, you would end up with a default of nano again

The two variables where in use long before files named *.desktop or mime* even existed.
(And the impressive thing is: they were actually used as a common standard.)
In Ubuntu, the system default seems to be set with sudo update-alternatives --config editor. It shows a menu to change the current association.

See section ENVIRONMENT in man visudo:

  VISUAL           Invoked by visudo as the editor to use

  EDITOR           Used by visudo if VISUAL is not set

If you never plan to use nano, you can also simply remove it. Then the system will use vi/vim as the default.

sudo apt-get purge nano

I know it is not the official answer, but it is one of the first commands for me after installing Ubuntu.

As described in this answer, add

Defaults editor=/path/to/editor

to the sudoers file.

Note: this will only work if the file being edited contains the Defaults editor=/path/to/editor line or includes a file that contains it.

For example: visudo -f /etc/sudoers.d/my_sudoers_extension will default to Nano.