This is what I see that might help:
The Dolphin Broswer is now also being published by Dolphin Browser (go figure) and I also noticed that instead of providing an update file they just uploaded a new version as a new app instead.
Maybe check and see that you have the newest version, by the actual Dolphin Browser developers. Looks like it is compatable with both devices too. That way you don't have to worry about syncing 2 different apps with two devices.
Suggest you try using the B&N version of Dolphin on your mother's Nook Tablet and use the Google Play version for your HD+. That way B&N won't try to change/delete the version on your HD+ and it will leave the version on the Nook Tablet alone.
I found out that the BN version of Dolphin did accidentally got sent to the Nook Cloud after all when my mother asked to look at my HD+ for a moment. Apparently, it needs to be active on BOTH tablets for it to stay put on her Nook. Thanks a lot Barnes! You've found yet another way to force unwanted stuff on your customers!