This is an issue I just solved and I share my solution here in case someone runs into a similar one.
The latest Python version available on apt is 3.5 for Linux Mint 18.3. I updated to Python3.7 via the following method (adapted from this and this):
Add a private repository
>>> sudo add-apt-repository ppa:deadsnakes/ppa
>>> sudo apt-get update
>>> sudo apt-get install python3.7
Make Python3 point to Python3.7 (or whatever version you've updated to)
>>> sudo update-alternatives --install /usr/bin/python3 python3 /usr/bin/python3.7 1
>>> sudo update-alternatives --config python3
Check the installation
>>> stat $(which python3)
>>> stat $(which python3.5)
Make pip point to the new Python version
>>> python3.7 -m pip install pip
In case someone follows this method: a large part of Mint relies on /usr/bin/python, which points to Python2.7, so don't change where this points to to avoid breaking the system, but python3 should be safe.
After that I noticed that mint wasn't working. When running it from the CLI, I got messages about missing packages: gi, pycurl, apturl, etc. These are not available from pip and need to be installed from apt. I tried to adapt this solution by copying and renaming the relevant packages into my local /home/[me]/.local/lib/python3.7/site-packages, but this is tedious.
It's not problematic in itself, as mintupdate is only a GUI for sudo apt-get upgrade
.