Can I calculate how much time it will take for an electrical signal to propagate through a TTL circuit?
I'd like to be able to calculate how much time it takes for (electricity? signal?) to move through a particular part/section of a circuit within a computer architecture...


That is what the "static timing analyser" of a synthesis tools does. The synthesis tools will adapt the used gates to meet the giving timing requirements. In that it will select the optimum gates, chose from various adders, do register multiplication, register balancing and use various other strategies to make the circuit meet timing. In that it will also estimate the 'wire load' depending on the area of the circuit. 

This task has moved away from 'humans' as it is way, way, too complex to do manually. 

The worth of a digital VLSI designer nowadays lies in the experience she/he has gained over the years to have a 'feel' for how the logic/RTL should look like to meet the required clock frequency.