DreamingInsanity:
Say if I had 20m of WS2812 LED strip, each with 60 leds/m (so 1200 leds in total), would the Arduino Uno struggle to drive this?
Yes.
DreamingInsanity:
Power isn't going to be an issue
Wanna bet?
1200 LEDs at 60 mA each full brightness is 72 Amps. That would be four of your 20 A supplies.
DreamingInsanity:
I will use one of those 20A led driver supplies. I will probably buy them in strips of 5m and run them like this:
so every new strip will be connected to the supply, which runs parallel to the strips.
You will need to connect the power to both ends of each 5 M strip and every metre along the strip. A clear error in your diagram is that you must connect the grounds from one strip to another wherever the data goes from one to the next and you should have the series resistor and supply capacitor at the input wherever it is feeding into a new power section given that you will use four supplies, one for each 5 M strip.
DreamingInsanity:
Will I be able to get these running smoothly?
Maybe one UNO per 5 M strip - I think 300 LEDs is supposed to be manageable but can be corrected on that. 
Nanos are more practical than UNOs but exactly the same specifications. 