Can't move servo with my Nano
Connect a multi meter in series with the servo connection from the uno to the servo and see how much current it draws when working (probably less than 20ma, definitely less than 40ma). Then connect it in series with the nano and see what happens. If you get no current then you're activating the wrong pin. 

If you see a current draw but the servo doesn't move then you need to use some sort of switch (transistor, FET, relay) to switch the servo. You could also compare the voltage and current read off the nano to the requirement in the datasheet for the servo. Again, this may lead you to needing a switch. 

Edit based on comments above... Also try your 5v directly to the servo, then add a forward biased diode in series which should simulate the voltage drop to the level of your nano (0.4-0.6v). If the servo doesn't work with the diode and does without then you can be pretty sure the issue is insufficient volts (which is likely given the rating of 4.8v you mentioned in comments).