Tested out the time setting function. The arduino is providing two frequencies, one per second and five per second. Both go into the 4066. The control pin for the one per second pulse is connected to a "voltage divider" with a switch on the low side. In the normal state, the button is not pushed, so the control pin for the one per second pulse is high, and the 4066 passes through the one per second pulse. The same control signal is also fed to the 74HC00 NAND. The other pin on the same NAND gate is connected to 5V. So in the normal state with no button push, the output of the NAND gate is low. This is connected to the control pin of the five per second pulse of the 4066. Because it's low, the 4066 gate blocks that signal.
When the button is pushed, the control pin of the one per second is now low, and the control pin for the five per second is high. The 4066 then lets through the five per second signal.
The 4066 outputs for the one and five per second are connected together, and to an LED that therefore flashes at one per second normally, and five per second when the button is pushed. I'll use this set up to allow the minutes to be set. Pushing the button will interrupt the normal one per minute pulse, and allow though a faster pulse rate. I'm not sure what speed is appropriate - either one per second or two per second seem like good candidates.