the last pic is not a R C circuit, even by knowing the winding resistance, you still have an inductive reactance to take into acount, the moment current begins to flow through the coil a countervoltage is set up that causes the current to lag the voltage between 0 and 90 degrees, as well as a capacitive reactance to add to that, so a time constant is not easily recognizable, what you really have is a series resonant circuit.
But will in actuallity function as a RC timing flasher.