No, the size of the wire is only important when the resistance gets significant. The 1/4" diameter for the receiving coil will give two problems:1. More turns will be needed, so the transformer ratio will reduce the current. The smaller diameter should be closer to the primary to get good coupling. The coupling is related to the ratio of the coil diameter to the spacing between coils.
I notice that I neglected to provide bias for the transistor, that error is corrected in this schematic. I simulated the circuit, there is about 10 mA current in the transmit coil, so the receive coil current will be about 3mA with perfect coupling. I think the LED will be rather dim. A 1" receive coil would work much better.