Why won't this rpi UPS circuit work?
First off, I think (hope?) that you drew your relay backwards. As shown, it would connect the Pi to the powerbank output when it is energized (when the charger is connected). If so, then...

You do not know what the circuit inside your charger or powerbank looks like. There could be lots of stuff going on inside there that could mess up this circuit. 

It is, for example, possible that the powerbank will output a voltage on the charging terminal when there is no charger connected and the current draw in the output is low. See how that could mess things up?

You could try adding a diode here...



..and see if that helps. If not, report back and we can try more things!

Also, while the 2 grounds on powerbank might be connected together, you really should not assume that. Better would be a double pole relay so each power should could have an isolated ground.