Ran through all of those steps (including the troubleshooting, and rebooting my machine).
Alfred Version: 4.0.5 
macOS Version: 10.15 (19A583)
Reponses to questions:
Are your snippets definitely ticked for auto-expansion?
Yes, I went through and unchecked & rechecked everything
What happens when you type your snippet shortcut? Does it disappear? Do you hear the beep for expansion, or an error beep? This will help establish at which stage the issue is happening and what may be misconfigured on your Mac
Does not disappear, no sound, don't find anything relevant in system.log or otherwise in console.app
What application are you testing this in? We recommend TextEdit as a basic native app to test in, to avoid the external factors brought in by non-native apps or website.
Safari, TextEdit, Xcode, Wunderlist, Pages. All of these used to work
When you show the Clipboard History, are items being copied to it?
Yes, it saves correctly and I can search my history, just does not paste when I select the item.
Do you use any hotkey modifiers (e.g. something where Cmd + V isn't used to paste), another clipboard manager that may be interfering, etc
No, not as far as I'm aware.