The chain does not reset if the ally fails to arrive. It only resets if the pokemon flees, your entire party dies or the last pokemon remaining on screen dies or gets captured. That's it. It's pretty foolproof. And expect allies to NOT arrive more often than not. You'll want a stalling tactic to avoid draining your PPs while this is happening. I recommend attempting to throw additional Adrenaline Orbs as it will not deplete your stock with the primary one still in effect. Additionally, make sure you shuffle pokemon so they don't run out of PPs and struggle themselves to death.
The last wrinkle is to be mindful of pokemon that call different allies than themselves as those allies typically won't call for assistance themselves and some are predatory and will end the chain immediately by killing off the pokemon you're chaining. In those instances, you'll need to kill off the stronger ally first so the chained pokemon can continue to call for additional help. I believe you can still chain allies this way for hidden abilities / shiny status but this will be easier said than done for some like Regenerator Mareanie and Prankster Sableye.