These days I encourage students to leave Bridge and Camera Raw behind and use Lightroom for all of their basic image editing needs (resorting to Photoshop when heavy guns are required). But if you do find a reason to use Bridge and Camera Raw, you’ll discover that Lightroom does not automatically see and show any Bridge keywording, rating or other metadata changes, or Camera Raw image adjustment changes that you have made.
To update your images in Lightroom with any changes you have made in Bridge or Camera Raw, select them, right-click and choose Metadata>Read Metadata from File. This will read the XMP file that Bridge and Camera Raw have written the changes to. Of course if you have since made changes in Lightroom that haven’t been written to XMP, this action will overwrite your recent changes. All the more reason to use only one program.