the final destination of these photos are to be displayed on google photo which only support sorting by timestamp.
Your issue needs a lot more detail to understand what's going on. Google Photos sorts by the capture date stored in the photo's metadata, not the time when the photo's file was created (e.g. by exporting from LR).
There are many "timestamps" associated with a photo and discussing metadata requires precision or else our heads will explode. The relevant ones for this discussion are:
- Capture Date -- when the shutter was pressed. Stored in several industry-standard metadata fields.
- Date Created -- when the operating system or camera firmware created the file containing the photo.
- Date Modified -- when the operating system or camera firmware last modified the file containing the photo.
For a photo file created in the camera, the latter two could be the same or very close to Capture Date embedded in the metadata. But as various apps modify the photo's file and after export from LR (which creates new files), they'll be much newer. Since LR exports files concurrently, there's no relationship between Capture Date and Date Created/Modified of the exported files.
I just retested Google Photos, and it sorts uploaded photos by Capture Date, not the Date Created or Date Modified. However, if a photo is missing Capture Date in its metadata, it will use the Date Modified as its notion of capture date.
So it sounds like your exported photos are missing Capture Date in their metadata. Most likely this is because you've chosen an option in LR's Export window that excludes it. Try exporting with the Export option Metadata > Include: All Metadata and retest the sorting in Google Photos.
If the photos are still misordered in Google Photos, that indicates that the photos in your LR catalog are missing capture dates (e.g. because they're scans or they come from very old digital cameras). If you get this far, I can give precise troubleshooting steps to diagnose and correct that.