After a good half an hour of confusion at why both my work laptop and home arch box are now seemingly refusing to connect to Github, this gem of a StackOverflow answer solved the issue for me.
ssh: connect to host github.com port 22: Connection timed out
That's right, I just needed to add a remapping for github.com → ssh.github.com
I know, strange one, but the most important thing is it works!
Github suggest that you use the command
ssh -T firstname.lastname@example.org to test your connection. Slam a couple of v's in there to get
ssh -vvT email@example.com and you've got yourself a nice debuggable output from OpenSSL.
In that output I could see an IP address, which on navigating to the IP in the browser, it does indeed route to Github. So I knew it wasn't a local DNS issue, it was finding the right IP address.
Half an hour wasted, but a Git crisis has been averted! Thank you Mahan_F!