For the inside digital time readout. Have you built the display on a breadboard yet to see how it looks? As you have it with 3x5 LEDs, it will be extremely hard to read. To do it that way, you'd want more LEDs in each bar.
You can't wire the LEDs as you show in the schematic. First when you're multiplexing, the resistors have to be on the side that can have multiple LEDs on at a time, so as you show it, you're scanning the 5 cathodes and outputting the correct pattern of anodes each time. With 100 ohm resistors you're using 20-30mA per LED (depending on colour), which means the cathodes will have to sink 160-240mA and the Arduino is limited to 40mA per IO pin. You'll need driver transistors on the cathodes to do it that way.
Why are you having it so you can control each LED at all? For example, won't the 5 LEDs that make up your 10's hour digit always all be on or always all be off?
A MAX7219 LED driver chip will allow you to control 64 LEDs or 8 7-Segment LEDs from 3 arduino pins, and the code to use it will be a lot easier than what you're trying to do.
There's also pre-assembled 4-digit modules you can use available from sure electronics, adafruit, and lot of other places with the driver built in so you just need 3 IO pins to control them.
You can also consider a bar graph LED driver IC such as the LM3914 for your seconds circle.
Your idea of using 2 UNO's is horrible. If you were short on IO there are far better solutions; you can use an Arduino Mega which gives you 70 IO pins. A port expander IC such as the MAX7313 will give you 16 IO pins and you can connect up to 64 of them to your Arduino using only the 2 I2C pins.
In this case though you're not short on IO, just use the correct driver chips and you shouldn't need more than 6 IO for all the LEDs you've described.