16.04
The sound has been part of unity-greeter but Ubuntu MATE 16.04 uses lightdm-gtk-greeter.
I couldn't find a configuration option nor could I get a script to work... but I found another way. 
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Open LightDM GTK+ Greeter Settings (in System β Administration) and go to the Misc. tab.
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Assuming you do not use the on-screen reader (orca), replace
orcawith:/usr/bin/canberra-gtk-play --id=system-ready -
Underneath, set the drop down to say "Enabled at start" and then click Save.
Only slight quirk is that it plays a second or two when the login screen first appears. 
18.04-20.04
The greeter is now slick-greeter which was a fork of unity-greeter.
In 20.04 onwards, install the ubuntu-sounds package.
In the terminal, edit this file as root, create it if it doesn't exist.
sudo nano /etc/lightdm/slick-greeter.conf
Paste this inside:
[Greeter]
play-ready-sound=/usr/share/sounds/ubuntu/stereo/system-ready.ogg
Press CTRL + X then Y to save.
This file will be overwritten if you use the Login Window tool in future!
21.04+
slick-greeter is now artica-greeter, which is a fork lead by the Ayatana project for distro-agnostic indicators.
Install the ubuntu-sounds package:
sudo apt install ubuntu-sounds
Similarly to 20.04:
sudo nano /etc/lightdm/lightdm.conf.d/90-arctica-greeter.conf
Add to the bottom (or any existing [Greeter]):
[Greeter]
play-ready-sound=/usr/share/sounds/ubuntu/stereo/system-ready.ogg
Press CTRL + X then Y to save.