If you reimport an asset with a different normal import method it won't update the build settings for that mesh. For example if you were import with computing tangents then set the mesh to import tangents then reimport it will still be set to Recompute tangents in the build settings.
The import settings do keep their changes and this can be a little misleading. This is causing users to think they need to remove the mesh and reimport it completely fresh into the mesh bypassing the built in reimporter.
If you uncheck "Recompute Tangents" in the build settings you get the desired results with the imported tangents.
The Tangents are still being recomputed in the build settings however the Normal Import Method has been updated in the import settings (see screenshots for additional clarification).
The Tangents are imported
There's no existing public thread on this issue, so head over toAnswerHub just mention UE-37520 in the post.