Voltage propagation along a wire
My understanding is that you want to feed a logic signal to two separate logic circuits with a delay in the order of pico-seconds.

Your proposal of using a single wire with two different taps makes sense. The #2 point will receive the signal with a certain delay like you expect. The problem is that the signal will reflect at the end of the wire and come back to node #2 and #1. Depending on the termination on the driver it will also reflect on the left side and come back to #1 and #2. The reflections will continue multiple times. What you'll actually see on #1 and #2 will be the superposition of all these reflections, which most certainly is not what you expect to have.

In order for your 1-wire approach to work, you need a few modifications. First, instead of a single wire, you need an extra wire for the ground return. I recommend you use a twisted pair. Then you need to add a resistor at the end of the wire to work as a termination load. Its value needs to be the same as the characteristic impedance of the transmission line implemented as a twisted pair. This will prevent reflections on the end of the wire pair. You don't have to worry about reflections on the driver side if you don't have reflections on the load side. Depending on the output resistance of your driver, the signal you will get on #1 and #2 may have lower amplitude than what you need (think about a voltage divider between the output impedance of the driver and the characteristic impedance of the transmission line). To mitigate this you may use several drivers in parallel to lower the source impedance or use different supply voltages.

Another option is to have two identical drivers with the same input, the first connected to a wire leading to #1 and the second longer wire leading to #2. The difference in length between the wires will determine the delay you are looking for. You also need a separate ground return for each connection like the single-wire solution. In this case, you will also have reflections at the end of the wire, since logic gate inputs are typically high impedance. However this reflection will not distort the signal because you are tapping right at the reflection point. Reflections on the driver side will cause you trouble, so you'll have to have matching on that side instead. The idea is to add a series resistor close to the driver so that this external resistance added to the source impedance of the driver equals the characteristic impedance of the twisted pair. One advantage of this approach is that you don't suffer from amplitude attenuation at nodes #1 and #2 (the traveling signal amplitude is half the supply voltage due to the source matching, but doubles in amplitude when reaches the end due to the high impedance reflection).