You first need to install gcc-7 and g++-7 since CUDA 10 only works with version 7. In case you have other gcc/g++ versions installed - the default on Ubuntu 19.10 should be 9 - you should do something like the following:
sudo apt-get install gcc-7 g++-7
sudo update-alternatives --remove-all gcc
sudo update-alternatives --remove-all g++
sudo update-alternatives --install /usr/bin/gcc gcc /usr/bin/gcc-7 50
sudo update-alternatives --install /usr/bin/g++ g++ /usr/bin/g++-7 50
sudo update-alternatives --config gcc
sudo update-alternatives --config g++
Detailed explanation for this step can be found here.
Now for installing NVIDIA drivers, CUDA toolkit, and cuDNN:
Following this post (that you already mentioned you tried), I recently managed to install CUDA 10.1 and TensorFlow 2.1.0 with GPU support for a machine running on Ubuntu 19.10 with 4 RTX 2070S GPUs. I used sudo apt-get install nvidia-driver-440
for installing the drivers (I was working on a clean install of Ubuntu so there were no previously installed drivers, but you can try sudo apt-get remove --purge '^nvidia-.*'
to remove existing drivers) and after that for cuDNN I followed instructions here but of course for cuDNN-10.1. I used this tarball instead of the debian package.
Make sure you pay attention to versions. TensorFlow 2.1.0 DOES NOT work with CUDA 10.2 - you should use 10.1. The same goes for cuDNN.