In such a situation, I poked at some math in the context of action saving with Productivity procs and got this:
As it stands if Snorri devoted 1 action himself and the apprentice actions into it as a baseline that would mean 5+2(Snorri)+2(apprentices) Progress on the first turn for 9 progress and then once Fjolla and Dolgi leave we'd make 5+2(Snorri)+1(Snerra) for 8 Progress a Turn and then Snerra we can assume leaves on turn 15, removing all apprentice actions and leaving it at 7 progress a turn.
We'd make progress in this way, 9 (Turn 12)+8 (Turn 13)+8 (Turn 14)+7 (Turn 15) = 32 on Turn 15. That's a baseline of four turns to complete it, technically five because it would complete at the end of Turn 15/start of Turn 16 and we couldn't do anything with it until Turn 16. That's one thing, and overall it would take 4 AP on our part and 4 from our apprentices to do it.
The defenses action would likely unlock and for the sake of comparison lets assume it takes as much as Defense In Depth Part 3 with 4 actions.
Totaled together that's 8 of our AP as a baseline, 4 from our apprentices.
If we slammed it and did all in on it every turn until it was complete then:
4 actions into it from Snorri is 6 actions due to Productivity which nets 12 progress, plus 2 from our Apprentices, plus 5 progress for 19 Progress total. That puts it at 11 Progress left on Turn 13.
On turn 13 if we assume we don't have our senior apprentices the count would be 12 progress from us, plus 1 from Snerra, plus 5 progress for 18 Progress. We would complete it with 7 progress overflow. If you make another assumption here and convert the Progress into AP that would be 3.5 AP.
That's not exactly a very helpful result, but it gives an idea on what that 7 progress overflow means in the context of both projects. The important part is that Soulcake was pretty clear that by doing something like this the defenses would be automatically completed. Going all in means devoting 8 of our AP, and then not needing to spend the assumed 4 AP on the defense request. If it didn't we'd need to spend 12 AP, but since it does we only spend 8 of our AP. The same as the baseline. If you count Apprentice AP that's 11 total, slightly less than the baseline.
Here is what applies to doing the Rune Metal this turn and then going all in until we complete it:
On turn 12 it gets 2 progress from apprentice actions, 5 from the Hold, for 7 Progress leaving it at 23. Turn 13 is 12 progress from Snorri, 1 from Snerra, 5 from the Hold for 18 Progress total. That leaves it at 5 Progress remaining. We have two options here.
We can do nothing on turn 14 and it auto completes. We only spend 4 of our actions and 3 of our apprentice actions. Whether this completes the defenses as well is uncertain. I would assume no, so based on my action count assumption the defense action would require 4 more AP. To finish that we would only need to actually spend 2 AP. That leaves us at spending 6 AP and 3 Apprentice actions, better than both actions above.
We can go all in again on turn 14. We spend 8 of our actions and 4 of our apprentice actions in total across Turn 13 and 14. This gives 13 overflow (18-5) on Progress. This completes the defenses as well, so we don't need to spend the 4 AP on it. That leaves us with 8 of our actions spent and 3 apprentice actions. Its not the AP lowest one at 8 of our AP and 4 of our apprentices, but time wise its the same as the one above, and ensures the defenses are completed.
Where the benefits show up are in reducing the amount of turns and thus Random Event Rolls we have to deal with until the Underway and the defenses are complete from five turns to two and in a more far reaching mechanical consequence means our Request category is only filled with the Banner request and the write in template. This effectively saves project space, and pushes the focus towards Research which would be the largest category.
Tl;dr the Rune Metal first option basically means we can end up spending the least amount of our personal AP on the Underway , which makes intuitive sense. Depending on what we do on turns 13 and 14 it can also be very fast completion, and leave us looking at an action list that is primarily research related. The all in first until completion route of Underway Ho! costs more of our personal AP, gets it done one turn faster, and leaves us with a list that is primarily research related.
Effectively a mostly intuitive trade of personal effort vs time it takes.
So maybe you could use Adamant to make the rune more easily, or more properly. So there's a horrible bottleneck in the form of a catch-22 almost. Prevented from being a full catch-22 by the fact that you can make Adamant without already having Adamant.
