You've seen how it's easier to change between the "C" and the "F" on the ukulele if you use your ring finger to play the "C".
Now try going from the "C" to the "G7" and back. Your ring finger just moves one fret, and stays on the same string. If you were using one of your more dominant fingers to play the "C", getting to the "G7" would be quite a process!
Finally, try the "G7" to the "F" and back. Your index finger stays put!
Take a moment, and just move from one chord to the next, around the circle - don't strum - and pay close attention to how your fingers get from one chord to the next. Soon, you'll be jumping back and forth by instinct, without thinking about it at all - that whole brain re-wiring itself thing again - so it will serve you well to play with it, and ponder how simple you can make the motions, before it becomes second nature.