This doesn’t work for me in version 5.2 and after, but if you’re using 5.0, it should work:
You need to use the Custom Properties on your mesh in Blender. Basically, you need to create a Custom Property with the perfectly matching name for every shape key you want to animate, and keyframe it in the NLA along with everything else. This won’t affect the actual shape keys (you can do that with drivers if you wish), but when you import the FBX into Unreal, it will convert the Custom Data properties into curves for any matching Morph Targets.
NOTE: You may need to tweak your import/export settings with Unreal to ensure it reads your custom data.
Also, I’m not sure if the “not working in UE5.2+” thing is a bug, or just a software problem on my side, because I’m using an older version of Blender. Hope that helps!