So it can be fairly established that this is infringement of copyright, but I'm unsure how to handle this. I've considered making a PR to the project that forked the other projects and adding the licenses/copyright (so that it can be merged to the "second" fork with permission of the person that added code) but wouldn't I have to get permission from the people that contributed to the "first" fork?
I've found it's helpful to be concrete, so here's my scenario: Alice, Bob, Carol, and Dave all write individual pieces of code de novo, and publish them under a variety of mutually-compatible free licences. Emily combines them all with some code of her own into a project which she publishes without any licence/copyright details. Frank picks this code up, further modifies it, and publishes the resulting code. You ask
So it can be fairly established that this is infringement of copyright, but I'm unsure how to handle this.
The principal offender here, it seems to me, is Emily. Even if the original four works (A-D) were published under permissive free licences, there is a requirement in nearly all cases to preserve copyright notices and licence texts. If any of A-D was published under a copyleft licence, then in addition the whole of E would need to be so published. So Emily's done a Bad Thing.
The second offender here is Frank. Since Emily's code contains no licensing information at all, Frank has no permission to copy it, much less to modify it and distribute the modified code. If you're ever in Frank's position, do not do this - we have other questions here from people who founded projects on uncertain copyright foundations, and caused themselves trouble further down the line.
I understand you're trying to work with Emily's codebase, to bring it retrospectively into compliance, so that Frank can rebase off it onto a better footing. That's a pretty good idea, but it may be a lot of work.
wouldn't I have to get permission from the people that contributed to the "first" fork
Assuming you mean Emily's code, are you saying there were other contributors besides Emily (and implicitly, Alice, Bob, Carol, and Dave). If there were, you're likely right: it's clear Emily has taken a very loose approach to copyright, so the chance of her having had a good CLA in place for those other contributors is nearly zero. If this is the case, the job is probably too big too tackle; find another project to contribute to.
However, the only people whose permissions you won't need are Alice, Bob, Carol, and Dave. They have already given permission for the re-use of their work, by using free licences. If Emily's codebase can come into compliance with the conditions of those licences, their permission has already been given.