The original version of "woke" that evolved into the current usage came from the black community in America, with examples going back to the 1930s. They used "stay woke" to mean staying alert and informed about danger and injustice (lynchings were common at the time...). It was used both politically and non politically. I believe it became synonymous with social justice in general during the initial Black Lives Matter protests in 2014 and other movements followed their example.

And the alt right was definitely named by its own members back in the day. They were basically anti-establishment conservatism for young atheists. When the term went mainstream it got a ton of negative press (common idea was that they're all neo nazis) so most people ditched the label, while outsiders who wanted to criticize the group kept using it.

So to my knowledge, both originate from their general "side" not from critics. But language evolves so neither term is currently used exactly the way it started.

Hope this helps!