Are they altium projects already? If so, start by creating an outjob file that exports a BOM in a suitable format, and see if there's enough data in a consistent enough format in the projects to get a BOM out of each. You can add a single outjob file and add it to all of the projects so you don't have to manage a bunch of copies, or you can copy the file into each project folder if it needs to be adjusted for each project. You might be able to write a user script that takes a bunch of projects and runs an outjob against each one, but that's probably only worthwhile if you have a LOT of projects.
If there isn't enough data in the project to create a useful BOM, then you will have to add it. This will probably be a lot of manual work no matter what. If the parts all target the same libraries maybe you can add the necessary information to the library component, but you will need to be very careful to not accidentally lose information in the process, like if someone manually edited part parameters after adding a part to the schematic. You will probably need to do a lot of manual verification.
Do you really need to do all of the projects at once, or can you go back and do them one at a time as needed?