Accessing EXIF data could vary somewhat depending on the image file format. I discovered many open source programs available to read and edit EXIF data for a large variety of file formats. The issue you're going to have is that none of them I found will run on a Java based server. Many will run in a Windows environment and there are some that will run in Linux or on a Mac. I believe I may have seen one that could also be hosted on AWS, but I won't say for certain.
I remember seeing some that ran using either Perl or Python making Heroku a possibility to host this service for yourself. There are sites on the web that can also read this data for you as a service (could be heavy on your API usage). You'd need to make a call-out to send the files to them so they could read the data and return the results to you.
Here are a few search results that you may find helpful to you: Free Open Source Metadata editors Software - SourceForge, batch EXIF edit - SourceForge, and EXIFTool GitHub. Just examining the code might allow you to do some of this in Apex. Whether your use case for this will be enough to justify the effort is something you'll need to determine. If your organization can't justify it, you might want to consider creating an open source project for this on GitHub.