In order to migrate to the latest Python release before the Anaconda Download packages haven been upgraded to it (in my case Python 3.8), I had to first upgrade my Conda installation:

conda update -n base -c defaults conda

And then create a new Python 3.8 environment:

 conda create -n py38 python=3.8

Unfortunately, a couple of packages are not yet there, so installing the full Anaconda package does not work. So the solution was to update conda and install packages manually:

conda install ipython
conda install numpy
conda install scipy
conda install matplotlib
conda install pandas
conda install jupyter
conda install astropy
conda install scikit-learn
conda install seaborn
conda install pillow

And now I can (hopefully) use the improved performance of Python 3.8 (benchmarks coming soon).

Packages not available through Conda still have to be added through PIP:

pip install --upgrade photutils
pip install --upgrade montage-wrapper
pip install --upgrade emcee
pip install --upgrade pymc
pip install --upgrade glueviz
pip install --upgrade aplpy
pip install --upgrade fits_tools
pip install --upgrade healpy
pip install --upgrade uncertainties
pip install --upgrade pysqlite2