Yes
Each Effect Happens Simultaneously
Hitting and dealing damage happen at the same time rather than sequentially, as described under the core rules for damage as well as under the rules for Strikes, but most broadly under the generic rules for Checks:
In the midst of combat, you attempt checks to determine if you can damage your foe with weapons, spells, or alchemical concoctions. On a successful check, you hit and deal damage.
Roll the attack roll for the weapon or unarmed attack you are using, and compare the result to the target creature's AC to determine the effect.
Pathfinder has many types of checks, from skill checks to attack rolls to saving throws, but they all follow these basic steps.
- Roll a d20 and identify the modifiers, bonuses, and penalties that apply.
- Calculate the result.
- Compare the result to the difficulty class (DC).
- Determine the degree of success and the effect.
Both of these runes' effects and the standard effect of a Strike are part of this final step resolving the attack roll. And as noted under @Ifusaso's answer, there's no mention in the rules of ordering effects that occur simultaneously.
It's also worth highlighting that the effect from bloodthirsty applies to a target that is bleeding when hit rather than one that was bleeding before being hit.
Bloodthirsty Wounding Weapon
So for a Strike in this case, the effect on a critically successful check for the attack roll is that:
- You hit and deal double damage
- You deal an extra 1d6 persistent bleed damage which is also doubled as damage.
- Your target becomes drained 1 if they are taking persistent bleed damage.
As each of these effects happen at the same time, the creature is bleeding when they are hit by the weapon and bloodthirsty would trigger to make the target drained 1.