While not necessary, creating a production circuit will free up your Uno for another project. Theoretically it means you'll also be able to create a smaller board, since you won't require a USB connection or other components that aren't used by the Cwik Clock! How small of a production board you can make boils down to your ability to solder in tight areas, and the way your PCB is constructed.
The schematic for the Arduino boards are available for you to download and use (please take a moment to thank the folks in Italy for making the hardware open source too!), but I've opted for a far lazier approach; using "Setting up an Arduino on a breadboard
" guide written by Carlyn Maw and updated by Rory Nugent as my guide (please take a moment to thank Carlyn and Rory!). By following along until the end of step 3 (but using our program on the ATmega), you should be able to replicate the circuit made in the previous step without your Arduino Uno.
It's up to you to determine how to transform that circuit into something more permanent. I decided to go with a type of perfboard, using a knife to cut the copper lines when needed. Some other options are to etch your own circuit, or have a company print you your circuit.