On the Galaxy client, it tells me my game version is 1.31, but when I load up my game it's in v1.02 and my save file is, as well. How do I update my game?

Do you have updates blocked in galaxy?For some reason or another the updates are being blocked.

It sounds like that's my problem. If so, how do I stop blocking the updates?

GOG Galaxy -> WITCHER 3 : WILD HUNT, THE -> MORE(this button located near another button "PLAY") -> SETTINGS -> UPDATES -> set value of rudiobutton to "Automaticaly". But I think that is not your problem. Anyway, good luck! ;)

Open galaxy, click on Witcher 3, click on "More", click "settings". If the settings you want are already ticked try ticking the other option then reticking the one you want. It can take a while sometimes for downloads to start sometimes.

Edit: After checking/ applying those settings if a download doesn't start or nothing changes you can also try verify/repair in the "Manage instalation" section.

If the updates are not blocked on galaxy,it could be your anti virus stopping them.

The options "Verify / Repair" and "Import Folder" are greyed out for me. I installed the game from disc rather than download, if that's relevant at all.

That could be relevant, maybe Galaxy isn't recognizing where your game is installed for some reason.

Try this: On the top left of the Galaxy client you'll see a "gog.com" icon with a menu arrow next to it. Open that and select scan/ import folders from that list and point it towards your Witcher 3 folder. If this doesn't work try turning your antivirus off and try it again. (I've had issues with antivirus and gog before-- if turning off the antivirus works then you'll need to add galaxy and your game folder to it's exceptions list.) If none of this works you'll need to contact GOG support, methinks.