Force Chromium to use Dolphin File Manager to save a download

Install plasma-browser-integration

sudo apt install plasma-browser-integration

Then install the plasma integration extension, and restart Chromium.

Image: Qt file manager widget in action in Kubuntu 20.04, with all bookmarked folders.

I have verified that this answer works for apt version of chromium and also firefox (the same plasma-browser-integration package and the same Plasma Integration add on). OP reported that it does not work with the snap version of Chromium, which Ubuntu ships. One can either can install chromium without snap instead, or open a bug at Launchpad to let the Ubuntu's chromium packaging team know about this.

To remove the snap version of chromium, enter the command sudo snap remove chromium.