Is it against the law to aggregate and publish legally gathered, publically available information such as:
- Social media usernames
- Real names
- Date of Birth
- Screenshots of posts/comments
- Email address
It might be a copyright violation to reproduce content that they created. (This question/answer explains fair use.)
It would not be illegal in the US to publish the other information as long as you don't make any threats.
The UK is still in the EU and is expected to keep the GDPR after Brexit becomes effective.
Some of the examples you mention (such as "real names") are what the EU defines as personal data (GDPR Article 4, definition 1).
Under EU law, it is illegal to process (that includes collecting, storing and publishing) personal data unless the activity meets the criteria for Lawfulness of processing (GDPR Article 6).
In your case, consent is the only legal base for processing that may apply.
This means that doing this illegal in the UK, unless you first obtain consent from the data subject.
If the information is about users who are unaware of or did not authorize the publicity, then it might be illegal. That information might be legally gathered and publicly available but that doesn't mean the public availability was authorized by the people who shared it.
But we are still trying to figure out what property rights are with respect to data in U.S. law. This is currently being discussed in Carpenter v. US 16 S.Ct. 402 (2017).