We really need to put this into the FAQ list. It's been asked and answered so many times.
Here is the gist: In TO, the hand moves the thumb to wherever it has to go. If you look at the Taubman technique, this principle applies to everything, not just scales. The thumb is supposed to move very little, definitely not all the way under the palm and also not a lot in the vertical direction.
"Why is that so?", you ask? Good question, because, of course, you should never just believe what people tell you without asking for cogent reasons

And indeed, there are cogent reasons. Essentially, the vertical movement of the thumb is equivalent to the horizontal movement of the other fingers, and the movement of the thumb under the palm (horizontally) is equivalent to the up-and-down (vertical) movement of the other fingers (take a look). You know how difficult (and slow) it is to move your fingers horizontally, and it is just as difficult to move your thumb up and down. In addition, doing this a lot can lead to injury. So, the idea is not to over-use the thumb. Use the arm to bring the hand where it has to be in order to place the thumb, then use arm rotations to bring the thumb on the key and to depress the key. As said already, the thumb can move on its own, but it shouldn't move a lot.
When to use TU vs. TO? Some people never use TU. I would say, use TO for all situations where you don't have to use TU. Once TO is mastered, you will find that there are very few cases where you need TU (e.g. in legato playing without pedal).
Finally, you don't always have to move the hand to play with your thumb. Take your C major scale, e.g., when you move from E to F, after you played the E, raise your elbow (keep finger 3 on the E), so that finger 3 is almost straight up. At the same time, move your elbow outwards. This will rotate your entire hand around finger 3. The thumb will automatically end up over the F. You just have to drop the elbow, and you are actually already playing the F. Everything wonderfully legato, without moving the thumb at all! Neither TO, nor TU. Well done, beautiful! At higher speed, the motions will become smaller. Use a combination of shoulder, elbow and wrist to realize the same idea.
Now, let's go down, let's say from A to G. Play the A with your thumb and keep in on the A. Use the same motions as described above, to pivot your hand around the thumb to place finger 3 over the G. You get the idea. Make a habit of using your thumb this way and it will thank you forever.
Finally, finally: there are always exceptions. It is OK to occasionally use weird and even dangerous movements if they are required to achieve a certain effect. Just don't use them often, and don't practice them over and over like your scales.
And most important: have fun!