I have opened a AE template and tried playing, but it plays in slow motion. So I obviously have a problem.
It is definitely not obvious that you have a problem.
No matter what you have in your composition, AE has to cache a preview before it can play it back in real time. The more complicated your project is, the longer it is going to take for AE to render that cache.
Now, if that template uses After Effects' obsolete ray-traced renderer, having an NVIDIA card would help, but that is a very big if. Very few templates used AE's ray-traced renderer. In fact, very few people used it at all. It was a pretty blah feature when it was introduced and is now considered obsolete. Read this for more info about that. Heck, it was pretty much considered obsolete from the very next version of AE after it was introduced!
Other than that one, obsolete feature, the current version of AE doesn't use the GPU for much. More info on how AE uses the GPU here.
Just so you know, the next version of AE that comes out will use the GPU to accelerate some effects, but it is unrelated to the ray-traced renderer and the error message you're seeing. I got to see some of it in action at NAB last month. The acceleration in the next version will work on AMD cards.
Anyway, to answer your actual question, no; don't worry about the ray-traced renderer and its requirements. It has gone the way of the Dodo.
Thanks for taking the time in getting back to me Szalam, much appreciated.
You were right, I got mixed up with playback and rendering.
AE works just fine with my setup. So released that I don't need to spend any money on a GC.
You're welcome! And welcome back to AE. Have fun with it!