If you didn't use the installer script, you likely only installed the root pyenv package.
If it's missing make sure you followed #3 AND #4 (restart your shell) under Install pyenv above.
Git is used by pyenv, plus it also enables builds/installs of source branches, so you could install whatever 3.8 is right now, i.e. Install tools and headers needed to build CPythons (exotic Pythons like PyPy or Jython may have other dependencies). See all the versions you can install with pyenv install -list Install pyenv
It will generally work same day of a release because you don't need to wait for someone else to package it for Ubuntu.
Plus, the answer to the follow-up question "How do I install Python 3.7 using apt-get?" has the same answer: pyenv update pyenv install 3.7.0. Everything runs as your user, so you don't have to worry about messing up the Python used by Ubuntu itself. You can build and install a new (or old) version of Python by simply saying pyenv install 3.6.0. It doesn't use Aptitude, and does involve "building it yourself", but it's fully automated. I would recommend pyenv to solve your woes.