How does Theresa May resigning change the Brexit (and sterling) equation?

Trick question, it does not!

The pound gained some ground earlier as May announced her resignation but quickly erased those gains with cable rising to 1.2710 before falling back to 1.2670 levels now. Amid all the chaos, what does this mean for the pound now?

Trick question, it does not!
Essentially, the Brexit path does not change as seen from the equation above. Regardless of a new leader, the impasse still remains. Hence, where should the pound go from here?
To put it short, it has now turned into a bit of gamble on who will be the next prime minister. Given that BoJo is favourite – and that he is a Brexiteer – the pound is basically trading on the odds of him taking over. If he is still going to be favourite, expect further downside pressure as a no-deal outcome becomes a likely possibility.