Easiest way is to get a mega (or whatever the original code was written for). You will have to sacrifice most of the testing capabilities if you stay with the uno.
The lcd needs 7 pins. Leaves yo with 7 digital pins and the 6 analog pins for i/o. There might be a button which you need to subtract and for debugging you might want to keep the serial port free for debugging.
If you want to redesign the complete system based on the uno, consider the use of i2c or spi port expanders or of shift registers.
I did not look at the code as I'm using phone at this moment.