(Note: EE is a british phone service provider.)
I was wondering if I could use the CyanogenMod installer on my Samsung Galaxy S3? It's in the supported devices list.
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I can't think of any reason why you couldn't try it, because it should either work and install CyanogenMod or it won't manage to install anything. It doesn't always work with my SPH-L710 S3 despite my being able to flash with Odin just fine.
Just do not install the wrong firmware on your own. There are over ten different revisions of the S3, and going against what's showing on http://wiki.cyanogenmod.org/w/SGS3FAQ for which specific model you fall under (most are considered d2lte now, but not all!) will likely end in a bricked phone. The differences between the international and the primarily US-only versions are somewhat notorious for bricking the careless.