How to replace the System's Download folder with one in an external drive?

This is possible via symbolic link.

1. Move your existing data in ~/Downloads to the location you want, let's say /ExternalDrive/Downloads

2. Open Terminal and run:

   This will remove the Dowloads folder and you'll need to type in your password to do that (you won't see *** when entering, just hit enter after you finish).

   sudo rm -rf ~/Downloads
   

3. Create a symbolic link to your desired location:

   ln -s "/ExternalDrive/Downloads" ~/Downloads
   

And voila! Dowloads folder moved.

A more detailed version of this process and be found here.

It's also possible to create a link to the iCloud Download folder. Using this command:

ln -s /Users/yourusernamehere/Library/Mobile Documents/com~apple~CloudDocs/Downloads  ~/Downloads

Your question doesn't specify whether you're just interested in changing the default Downloads folder location for your web browsers, but since web browsers are the most common way that users download files from the internet, below are instructions on changing this location within Safari, Firefox and Chrome.

Safari

For Safari:

1. Go to Safari > Preferences (or just press command,)
2. Make sure you have the General icon (top left) selected
3. Find the File download location drop-down menu and select Other... from your list of choices
4. Navigate to and select your preferred Downloads folder location
5. Exit Preferences

Firefox

Very similar to Safari, in Firefox:

1. Go to Firefox > Preferences (or just press command,)
2. Make sure you have the General icon (top left) selected
3. Under the Downloads heading you'll see a radio button labelled as Save files to with a field next to it. Click on the Choose... button
4. Navigate to and select your preferred Downloads folder location
5. Exit Preferences

Chrome

Finally to Chrome:

1. Go to Chrome > Preferences (or just press command,)
2. Scroll to the bottom of the window and click on Advanced
3. Now scroll down further until you see the Downloads heading
4. Click on Change
5. Navigate to and select your preferred Downloads folder location
6. Exit Preferences

NOTE:

Similarly, others browsers and 3rd party downloaders also provide a setting in their preferences from which you can choose your Downloads folder location.