Magicka would ONLY govern how much magicka you have to spend.
Health would only govern how much HP you have.
Stamina would stay the same, except it would have no effect on how high your carrying-capacity is.
Athletics would effect how fast you move and how fast your attacks are, but it would also effect how high you can jump.
Agility would effect how quickly you change direction (side to side), and would also effect your ability to block/dodge (the higher your agility, the quicker you can raise your shield/sword to block, and the quicker you can dodge). Think of it as "reflexes" instead of agility if you must.
Strength would effect your carrying capacity, and it would also effect how much of an effect heavy armor has on your movement speed.
Intelligence would effect what level magical spells you can cast, and what level of weapon-attack techniques you would be able to perform. (As far as I know, there are no "weapon-attack techniques to learn in Skyrim, so Bethesda would have to add these. They could range from more/better killing-animations, to more effective/powerful melee attacks).
You know, I like this idea a lot. It seems to have a good balance.
The only flaw I notice (there may be more, I'm pretty tired right now

but maybe not. I guess I'm just used to having my sensitivity rather high. And for some reason, on Skyrim, I have to turn the sensitivity ALL the way up, or else my mouse moves at a snails pace. I'm not sure why it's like this.
Eh, I'm not so sure that would be a good idea. Unless there were some tweaks done to the amount of skills required to level up, where you need more to level up, considering that these "skills" would be used quiet frequently it seems.