Q: Apple Aperture/Photos transitioning to Adobe Bridge

The devil's in the details.

First, Aperture can write metadata like keywords, captions, etc into the files themselves. Some Aperture stuff, like folders-albums-projects, ratings, stacking, etc, doesn't get written to files. But you could substitute keywords for that (use a project name for a keyword, eg, or 1star for ratings, etc. Keywords are pretty universal so this helps transfer organization info to other platforms, or applications like Bridge, which can read that info.

Second, you then copy or move the images that have had the metadata written within to a new location, and  use Bridge to browse them. Bridge would see the keywords, etc. You could search and filter on those keywords. Filter say all the keywords of "1star" and then apply one star ratings to them. And so on.

Bridge is more of a browser compared to Aperture and Lightroom; it doesn't import stuff like they do. It also doesn't have an import plugin like Lr does. But even if you used Lr, note that it has no importer plugin for Photos. And in Photos it's much harder to manage metadata like yours, let alone write it to files. So the best bet is to use Aperture at work to do this.

Another way to go is to use Aperture Exporter at work; it requires Aperture however. See their website for details.

Others might be able to tell you how to get a copy of Aperture on your new machine; I'm unsure of how to do that. You could get your boss to splurge for Lr on your new computer, or simply get the trial version and run all the importer functions with the Aperture libraries and get most of the stuff out that way. Then just use Bridge going forware.

If the "even older" library was converted to Photos, then it faces the same problems. Even Lr won't work on that. You'd probably have to do a lot of manual keywording and such. But at least you could copy out all the originals from within the Photos Library for use in Bridge.

Caveat Emptor - the latest version of Adobe Bridge is incompatible with OSX El Capitan. Your version of Adobe Bridge will not be able to recognize any camera devices that you connect to a current OSX El Capitan platform. You will need to import your images using the built in OSX Image Capture. Then save your keeper RAW or NEF images to another folder that you can access with your company network version of Adobe Lightroom. Good luck.

I am on a new Mac though that does not have Aperture, only Apple Photos so is there any way that I can output the organizational structure from an Aperture file and import it into Bridge without having access to Aperture?

If you open an Aperture Library in Photos it will preserve most of the structure, edits,  and metadata, so you can export the photos to folders with added metadata. See this link:  How Photos handles content and metadata from iPhoto and Aperture - Apple Support

I am on a new Mac though that does not have Aperture,

If you had Aperture 3.6 on your previous Mac, purchased with your AppleID you could use that to download and install Aperture 3.6 on the new Mac from the Purchases history, the fourth tab of the App Store.This will only work, if Aperture is associated with your AppleID at the App Store. You cannot reinstall the DVD versions of Aperture on a new Mac.