As the title states, what problems should one expect. I've seen a few, and I'll note them in the self-answer, but if you've seen any others, I don't think this has been asked/inventoried before.
If you remove the python3 symlink and link it to python3.7, you can expect some oddities.
You won't have python3 -m pip until you reinstall
python3-pip in some way. Doing it through the package manager works.
apt_python won't be found, and reinstalling from the package doesn't work, but copying the .so file with a rename does:
sudo cp /usr/lib/python3/dist-packages/apt_pkg.cpython-35m-x86_64-linux-gnu.so /usr/lib/python3.7/apt_pkg.cpython-37m-x86_64-linux-gnu.so
gdbm won't be found, if you have some need for it, but copying the .so again works:
sudo ln -s /usr/lib/python3.5/lib-dynload/_gdbm.cpython-35m-x86_64-linux-gnu.so /usr/lib/python3.7/lib-dynload/_gdbm.cpython-37m-x86_64-linux-gnu.so