How does one estimate how long a part will take to be made? Is it purely by analogy (this part is similar to something we made earlier, therefore it should take about the same amount of time) or is it from fundamentals? (this part has x number of welds, faces to cut, etc.)
I used to work-with/design high complexity, low volume, high performance biomedical devices that made use of materials and processes similar to rocket components, so the following is based on my personal experience (your mileage may vary, more experienced engineers may point and laugh) over several years and projects.
You start by analyzing the part and your chosen production method. How many cuts/welds/coats/folds/etc does the part have? How are they orientated? Do you need to use more that one machine? How many machines? How can you move the part from one machine to the next? How many different tasks are involved in assembly?
You should end up with a highly detailed list of every task required to go from metal stock and components to finished part. Next you add a time to each step.
If the part is going to use subtractive manufacturing you hand your CAD files to a CNC programmer who computes tool paths based on the geometry of the part and comes back with the total time the part will need on the machine.
This is the time drawn from fundamentals you stated in the question; based purely on cuts/processes defining the part. Substitute the CNC machine for a 3D printer or any other large production machine for our particular part, the process is the same.
What is harder to estimate accurately is how much time outside the machines will be required. What may look a perfectly straightforward task often turns out to be unexpectedly complicated. You look at a sprung valve assembly and assume that it would take mere minutes to slot together. Then you actually try and it's impossible to hold both o-rings compressing the spring while holding the enclosure open with your other hand, and your hand better not slip or the spring will launch an o-ring across the lab, which you will then spend the next five minutes searching for on hands and knees.
Experience helps a lot (and sure would have helped me), but for novel devices/parts there's no substitute for actually doing it. Walking through the process step by step, literally walking the route the part would take across the factory floor, stopping at every station and performing (or simulating) every task, assembly, and process with your pre-production prototype (You do have one of those, right?). Even then your estimate will only be accurate if the production line is staffed with clones of yourself. For really accurate timings you'll need to try it with real manufacturing technicians, that's usually called "production" and you better hope you got the estimates right before that.
Most important is talking to the people who are actually going to make the part, manufacturing technicians and production engineers have invaluable experience which can allow them to make very accurate estimates on most parts of the production process.
(At some point you'll probably want to draw up standard times for the part and/or production processes, sometimes this is part of the pre-production time estimations, other times it's used during production to analyse and optimize processes and resource flow)
I'm no expert, and there's a lot more to read on the subject if you're interested:
Knowledge-based Estimation of Manufacturing Lead Time for Complex Engineered-to-order Products For more advanced methods.
Implementing Time Studies And The Development Of Standard Times. For more reading on Standard Times
NASA Cost Estimating Handbook For a Nasa/Spaceflight focused in depth cost guide, also includes a little on time estimations.