If you have a hero with Timeslicer and Heartseeker out and you attack an ally, can you choose to strike with Timeslicer first and then Heartseeker to get Heatseeker's bonus or do both weapons tap at the same time before damage is dealt?

Weapons are swung during the defend window.  When the strike resolves, it adds the ATK of the weapon to your hero.  No damage is actually done at this point. 

After the defend window closes, you move to combat conclusion where the attacker and defender trade damage equal to their ATK.  This process will create 1 packet of damage with the ATK of your hero (and 1 packet of damage equal to the ATK of the defender).  Whether you swing with 1 or 2 weapons only affects the amount of ATK your hero will have.  You still only deal damage once.

So, to answer your question, no.  There is no way to get heartseeker's bonus by striking with timeslicer unless they are swung in separate combats.