The pressure allowing the stick to move when pushed at one end will move at the speed of sound in that material.
Imagine the rod is a slinky. Understand springs (Hooke's Law, etc.) and you understand solids in general: they are just orders of magnitude stiffer. It might be more philosophically correct to note that a spring is just like any solid rod, just wimpier. A wall resists pushing because it compresses by some invisible amount.
Imagine a short rod of steel the size of a pencil. On this scale you find it hard and unbending. But, it's really a sample of a product sold as rope! It comes in spools 10 feet in diameter. For a length of a bridge span, it wobbles and obviously ropelike.
If you have a steel I-beam a mile long, and you push one end hard enough to shove it, you will note a delay before the far end moves, and the length shortens like a piece of rubber, compressing first, and then the wave of compression moving to the other end where that eventually causes the far end to move.
Radio would be far faster than mechanical linkages.