Because the applications work in fundamentally different ways, it’s often necessary to think through multi-application edits and do them in a specific order. I have to deal with this between Lightroom Classic and regular Photoshop, and this is one way to think it through:
When the image is opened in the other application, check its metadata to make sure it made it that far. I don’t have Photoshop Elements, but in regular Photoshop that can be done by choosing File > File Info. If the metadata is missing, go back to Lightroom Classic and choose Metadata > Save Metadata to File to make sure the metadata is available to other applications. Then, when saving out of the other application, make sure it’s being written out in a way that includes metadata. For example, in regular Photoshop if you choose File > Save a Copy the metadata is included, but if Save for Web (Legacy) is used, it has to be set to include metadata or it will not be included.
It’s a lot of hoops to jump through, but metadata can be dropped at various points if some settings are not quite right.