When does an airliner switch from using the tiller to the rudder?
I flew for two 747 carriers that never bought new aircraft, and thus this answer applies to 747-100/200 aircraft as originally manufactured for a number of different airlines.

The tiller is active when the aircraft is on the ground and the nose gear is compressed regardless of groundspeed as I remember.

I've never heard of a pilot accidentally hitting it at high speed on the ground, but if they did it would scrub the front tires. It wouldn't appreciably turn the airplane.

Pilots did have to be wary of having their hand within the travel range of the tiller when a tug is being hooked up. Ground handlers would sometimes work the nose gear back and forth a bit, and that action is reflected in tiller movement. If they do it quickly, the tiller will snap quickly and a hand in the way will get hurt.

The nose gear is relatively weak compared to the rudder, the brakes, or the engines insofar as its ability to turn the aircraft. To make turns off a runway onto a taxiway that is 90&deg; to the runway, you usually aim for a 10 knots or less groundspeed. You can do a little more depending on the runway surface and what your center of gravity is, but sometimes 10 knots can be too much. I once skidded the nose gear badly at Jeddah, Saudi Arabia at midday in the summer when I failed to account for the fact that hot asphalt is slippery.

To illustrate how poor tires are in preventing sideways scrubbing, consider the 747 body gear. 747s have body gear steering for tight turns, but you can dispatch the aircraft with inoperative body gear steering. You can still make tight turns, but not quite as tight, but you do it with engine power and just scrub the body gear around, all 8 tires with a lot of weight on them. You might have difficult doing that, though, if you c.g. is greater than 26.6% mac. That's where the wing gear is, and if you're aft of that (limit is 33% to 37% typically) you're got a lot more weight on the body gear. Body gear steering is disarmed for takeoff and landing though.

Barring a strong crosswind, the usual takeoff procedure was to not use the tiller unless you were doing a rolling takeoff, starting the power up before being aligned with the runway. Once aligned and moving the rudder will do it. Even in a strong crosswind, you can forget the tiller very soon in the takeoff roll.

On landing, no use of the tiller until down to taxi speed except maybe a little use for a strong crosswind just before reaching taxi speed.

Boeing offered a rudder pedal-nose gear steering link, but almost all 747s I flew did not have that. I flew a couple of airplanes that had it. It helped in a crosswind.