If I check this, what will happen to the existing images in the uploads directory (will they be re-organized?)
No. The old structure is preserved and new files are added to the chosen new structure
If not, can someone recommend a way/plugin to refile them in the new structure? I’d like to tidy up.
The problem you will run into with reorganizing them is that links to the files in posts and pages will not update if you are simply moving stuff around the server. An integrated solution is required. While there are plugins that purport to do file management that should help (or at least make this slightly less painful) I tend to think your best plan of attack would be to think through what kind of organization you need and for whom. Given your scenario above, it should be easy enough to placate the ISP by moving a chunk of files to a new subfolder and doing a database-level search and replace (or attempt to use the file management plugin linked above) and just keep things organized going forward. Depending on your setup and budget, you might also want to think about sending new uploads to S3 or other offsite storage option…