My base game is v 1.10 that is required for the expansion. However, every time I try to install the expansion it says that my base game is not the required version! Please help... Do I have to uninstall the Witcher 3 and redownload everything?

Are you using GOG Galaxy? If yes does Galaxy thinks that the game is updated?
You could try this: Deinstall Galaxy and install it again. Import Witcher 3 to Galaxy and hope it will recognize this time that Witcher 3 is version 1.10 now.

== IMPORT ALREADY INSTALLED GAMES TO GALAXY ==

1. For GOG games installed using installers from the last few months (so called Galaxy-compatible installers)
Click the "+" button on top of the sidebar and select "scan and import folders" - it will find all compatible games within that folder and add them to the Client.

2. For remaining GOG game INSTALLATIONS
Find the game in the Library (click on the image of the game), then click the More button and select "Manage" -> "Import folder" and point the folder selector into the folder where that game is installed.
Please be aware that Client will accept any folder pointed to and will overwrite files inside it (this can for example wipe some mods or damage other apps in case of pointing to a wrong folder) and will update the game to the newest version (current default language English).
Source: http://www.gog.com/forum/general/gog_galaxy_beta_2/post1

This worked here for another person:
http://www.gog.com/forum/the_witcher_3_wild_hunt/gog_galaxy_patch_version_not_match_with_the_game_version/post1

If this doesn`t help you could try to repair your game with Galaxy.

I have the GOG version, downloaded patch 1.10 , my game says its 1.10 too (ingame in pause menu)

But if i want to update to HoS it says This expansion requires patch 1.10......

i found a solution (GOG Version without GOG Galaxy ! )

i tried to re-install the game, but it was not possible to Update to 1.10, because my game Version is the 1.0 and require the Day One Patch.

I downloaded the 1.06 Base Game -> Update 1.10 > Language Patch -> DLCs -> Heart of Stone Expansion

Now is everything okay!