The cooldown at Q for ranks 1-5 is 10, 9, 8, 7, 6.
The cooldown for W for ranks 1-5 is 7, 6, 5, 4, 3.
The DAMAGE on the Q spell scales better, however the total damage output of W is higher thanks to Sheen. Q can still proc sheen, but when both spells are maxed, W is hitting twice as often. So, when I play Jax, I'm typically playing a pretty passive laning phase and farming. I'm aiming for the Sheen. That's Jax's big powerspike, and maxing W first maximizes your damage out put with Sheen.
The reason to take Q at level one is because of the free mobility. It's a 10 second flash. If the enemy decides to duel you at level 1, just Leap Strike away. Jax isn't going to be all that great at dueling till 3 whether you take E or Q first. Meanwhile, taking Q first makes you better at trading.
And with Jax, I'm never ultra worried about "oh no, I had to flash" at level 1 because I have my Q. The Q can be used offensively or defensively during an invade, while E can only be used if I can stay close to my target. I may have to use flash anyway on an invade with E just to stun them and get the kill. I'd rather just have Q.
But Jax, like most tops, has to be in melee range to last hit anyway, so taking Q first means you can auto-attack an enemy and walk away, or you can Q-in to the enemy, auto-attack him, snag a last hit, then walk away.
If your FLASH cooldown is the reason you're doing anything on Jax, then you're playing the wrong champion.