@merrymegeden OK I can see in the DxDiag that something seems to be causing Origin to hang while trying to run, Usually allowing Origin through any anti-virus should resolve this.
Since it is still happening I would recommend backing up any save and mods you have to a folder on your desktop. Then clean uninstalling Origin and Sims. Next run something similar to CCleaner to clean your PC and registry and try to install both again and see if this keeps happening.
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