I'm not familiar with the Adobe Bridge product, but some products only apply "edit layers" to your photos, so that the original photos are left untouched. Opening them in those programs you would think the changes have been "saved", but other programs won't recognize the saved layers, so they only display the original photo.
Try to find another image editor that doesn't use layering, or look for an option to 'flatten' the image when you save or export.
You are right to be concerned with not wanting to recompress the photo again. Most people don't realize that the Jpeg image format is a lossy format, meaning it throws away and loses image data when it saves. That's why Jpeg is a very poor format for archiving history. (Its like making a Xerox copy of a Xerox, with each copy generation the quality gets worse and worse until eventually it becomes illegible). Always try to work with original works in a lossless format (tiff or png), and only save the final as a smaller jpeg for sharing. http://www.scantips.com/basics09.html