Fixed with compiling python from source + adding the usr/local/bin to PATH and updating alternatives for python.

I'm also using the deadsnakes PPA. Every version is installed separately, so you can switch versions on the fly. So to use any version installed from deadsnakes, you have to call it explicitely.

Copy$ python3.10 --version
Python 3.10.4

So finally I found out how to install Python 3.10 from sources

dependencies needed for building:

sudo apt install -y build-essential libssl-dev libbz2-dev libreadline-dev libsqlite3-dev wget curl llvm libffi-dev zlib1g-dev

Download and extract the sources

cd /opt
sudo wget https://www.python.org/ftp/python/3.10.12/Python-3.10.12.tgz
sudo tar -xvf Python-3.10.12.tgz

Compile and Install

cd Python-3.10.12
sudo ./configure --enable-optimizations
sudo make altinstall

The deadsnakes ppa does not contain 3.10 for Jammy (22.04). So it's working as designed, telling you that there's no 3.10 other than the one you have installed. There is 3.11 and 3.12 for Jammy. But I certainly wouldn't replace the system python with those.

You may want to consider using virtualenv's to isolate your work for different python versions.