WhatsApp doesn't prompt to restore from local storage
I had problems with this too, and today finally it worked.  I'm not sure which of the following additional steps did the trick, so I'll list both:


I copied the entire WhatsApp folder over from the old phone to the new one (apparently the asker already did this).
After installing WhatsApp, I gave the app all possible permissions before running it: under Settings/Apps, select WhatsApp, tap Permissions, and then select all permissions listed under "denied" and tap "allow" (you can revoke the permissions again after the backup is succesfully restored).


After doing this, running WhatsApp and confirming my phone number, I was finally prompted to restore the local backup. Before that, I had selected "give permission" in the popup from the screenshot above, then approved the permissions it asked for, but apparently this is only for the Google Drive restore. It looks like restoring a local backup only works if the permissions are already present?!