We're having a bit of a discrepancy in our campaign:

Jyn's "Quick Draw" states "use at the start of a hostile figure's activation. Interrupt to perform an attack...." but Szark in the campaign guide has a special rule stating "At the beginning of an activation pay 2 threat to heal."

Since Jyn is interrupting does the attack resolve before the imperial player has the opportunity to heal?

I would imagine it falls under the Timing rules on page 2 of the RRG:

"In a campaign, resolve mission rules first, followed by effects triggered by the Imperial player, then effects triggered by Rebel players"

We saw that but assumed that just as if you'd interrupt an action, that interrupt would resolve before the action, that if you interrupt an activation the interrupt would resolve before the activation.

Both are at the start of an activation, so the timing rules apply.

(Also, the word "Interrupt" is mostly an unnecessary word.)

The interrupt doesn't immediately take precedence, just because it says it's an interrupt.

Any things happening at the same time all follow the timing rules as mentioned above.

Played correctly, (and with dice luck) this mission can be a rebel victory after two activations (1 rebel, 1 imperial).

"Interrupt" usually just refers to something being done outside of a figure's own activation..