Since you do
UE4 opens UE4 native browser
User logins into Twitch,
user accepts/denies application
User is redirected to the specified redirect
You can use a custom URI Schemes in a redirect. (myapp://twitchredirect course myapp will be more named like your thing is called)
So if you UE4 project can register one, you can redirect to that.
A quick google suggests
The suggestion don’t look very Unreal-y, I know Electron (nodeJS) has a bunch of handlers built in to do it “easily”, rather than this registry soluition. As I don’t speak Unreal I don’t know what Unreal can do, but a custom URI schema is the way to go here I think, and should at least point you in the right direction