Installing the mod that bypasses the disc and running the game in compatibility mode for Windows XP SP2 fixed the problem for me. A modified Sims.exe to do this can be found here:
I have uploaded it to Google Drive, as the website I found it on was a bit sketchy. I scanned it with my anitvirus and it came up normal... Google Drive also scans anything you upload with it's antivirus, but regardless use at your own risk. Please note that this is not a link to a pirated version of the game, just a fixed Sims.exe that will allow the game to launch on Windows 10. I own the game and installed it on my computer via the installation disks. Obviously you will have to own the game too for this to work.
Instructions on getting it working:
1.) Make sure you already have the game installed on your computer.
2.) Go to :C > Program Files (x86) > Maxis > Sims. Rename the Sims.exe to sims.old. Make sure you keep this file in case the new one doesn't work.
3.) Download the file and move the new Sims.exe to the same folder (:C > Program Files (x86) > Maxis > Sims).
4.) Launch, run the game in compatibility mode if necessary.
5.) If it works, enjoy. If not, remove the new Sims.exe and change the old one to it's original name.
Hopefully this helps anyone who's spent hours searching the internet trying to find a solution to this issue.