Try a different pin from 13 on the Uno Rx. The led on that pin can sometimes effect its alternative uses.
I tried pins 9 & 8 and 2 & 3 (RX, TX). Neither combination worked.
Can you get the HC05 on the Uno to communicate with a phone app like Kai Morich's Serial Bluetooth Terminal?
I only have an iPhone and all the apps I tried were not able to find the HC05.