With iOS 13 Apple enabled the ability to download videos from your browser directly to your phone just by pressing on the download button from video management.
Press this button and you will be shown the following prompt:
This will download the file to your Files app.
⚠️Make sure you have this - from the App Store - Apple is the Developer ⚠️
You will be able to access the file from the browser download shortcut, or from the files app once it's done!
That's all there is to it!
If you run into issues, check that you are updated on iOS, Safari, and Files in your App Store and that you are online.