Compare the python and python3 symlinks in /etc/alternatives/ (e.g. with ls -lF /etc/alternatives/python*.

My guess is that python points to python3.8 while python3 points to python3.10. Or vice-versa.

Is there any particular reason why you want python to run 3.8 instead of 3.10? if not, run update-alternatives again and set it to either auto (preferred) or to 3.10 (you'll run into the same problem in future when 3.11 or 3.12 or whatever is the latest auto version).

Or just make sure that the alternatives symlinks for python and python3 point to the same version of python.

replace

[bash:~] $ sudo update-alternatives --install /usr/bin/python python \
/usr/bin/python2.7 2

[bash:~] $ sudo update-alternatives --install /usr/bin/python python \
/usr/bin/python3.5 3

with

[bash:~] $ sudo update-alternatives --install /usr/local/bin/python python \
/usr/bin/python2.7 2

[bash:~] $ sudo update-alternatives --install /usr/local/bin/python python \
/usr/bin/python3.5 3

e.g. installing into /usr/local/bin instead of /usr/bin.

and ensure the /usr/local/bin is before /usr/bin in PATH.

i.e.

[bash:~] $ echo $PATH
/usr/local/bin:/usr/bin:/bin

Ensure this always is the case by adding

export PATH=/usr/local/bin:$PATH

to the end of your ~/.bashrc file. Prefixing the PATH environment variable with custom bin folder such as /usr/local/bin or /opt/<some install>/bin is generally recommended to ensure that customizations are found before the default system ones.

Even if update-alternatives work, do not change the default version of python. It may break your system.

It may be better to use alias in bashrc

open $HOME/.bashrc in a text editor and append

alias python=python3.7

This will change the python version on your terminal only and should not affect system programs that depend on it.

I have answered my own question. First, I did have alternatives, but as Ubuntu defaults to Python2 (please Ubuntu change that -- the world has moved on), the command is sudo update-alternatives --config python3:

$ sudo update-alternatives --config python3
There are 2 choices for the alternative python3 (providing /usr/bin/python3).

  Selection    Path                Priority   Status
------------------------------------------------------------
* 0            /usr/bin/python3.8   2         auto mode
  1            /usr/bin/python3.6   1         manual mode
  2            /usr/bin/python3.8   2         manual mode

Note the line "providing /usr/bin/python3." That means the Python symlinks for Python 3 are in /usr/bin/python3. So I add the new one:

sudo update-alternatives --install /usr/bin/python3 python3 /usr/local/bin/python3.10 3

That makes Python 3.10 priority 3, the highest priority. Then:

$ python3 --version
Python 3.10.0   

I hope that helps someone in the future.