I think you and most answers make a logical error here. Since this is not a natural process, you can not actually approximate its duration or nature from biology despite it being based on biotech. All that depends on how the process and the related equipment has been designed. Which is limited by the limitations of the available technology:
Unfortunately the technology here is in the handwavium territory so it does not have any real limitations. You could perfectly reasonably say it is a process of "push button and wait five seconds for the bell to ring" as far as hard limits go.
However there are some reasonable assumptions we can make.
First detaching from the ship must work as an emergency procedure.
If something goes catastrophically wrong and the ship must be abandoned, the pilot must have a realistic chance to walk (or run) away. This means detachment can not rely on the ship being intact enough to grow a cocoon and a body and then transfer memories to the brain in the body after the crew triggers the process.
THis means that both the protective cocoon and the backup body must exist from the beginning, be constantly maintained at working condition by the cocoon and at all times contain a warm backup of the mind.
If the detachment is triggered either by the crew or by the ship systems collapsing the backup is simply cutoff from the cocoon which makes the cocoon and some sort of an emergency exit to pop open.
This is fast and simple and should take somewhere between five seconds and five minutes. Probably closer to the seconds as the body would probably be pumped full of drugs that instantly snap it fully awake and suppress inevitable confusion and disorientation.
Waiting while you grow up sucks, ask any teen ager
It makes no real sense (as noted in comments) to wait while you grow up to be a ship, if you need a human body anyway for emergency detachment and the growing up is handled by huge engineering marvel with handwavium technology.
Why would people actually do that?
Instead you should model the process on modern pilot training. The pilots are trained and the ship are built in parallel. This instantly makes the process twice as efficient. It also means that pilot training or ship construction failures do not instantly freeze the other part of the process.
How do you make the pilot?
Astrogation training is fine but modern airlines do not give some lectures on aviation and then put you responsible to flying hundreds of people safely. Instead simulators are used.
This would go doubly here since the pilots would need to directly and persistently connect to their ships. As such the necessary neural links would be created during the training process, the pilots would be trained using simulator cocoons that allow them get used to "being a ship instead of a human" in bite size steps.
This would also allow them to find out if they have issues not being a human anymore before being transformed into ridiculously expensive spaceship. And make the necessary adjustments over years of simulator training with constantly available counseling and even therapy. People funding all this would also appreciate only needing to built ships for people who are actually good at it in simulations.
Same would go in reverse. While being a ship the pilot would periodically, possibly subconsciously. run simulations of running and acting as a human. It would be embarrassing if you woke up as a monkey in an emergency that killed your real body and it turned out that you have forgotten how to run away.
These simulation might allow the ship to interact as a human with the crew in virtual reality but I am not sure if someone mentally adjusted to being a huge ship would want to interact with monkeys that mostly think of having sex and eating bananas. Simulated emergency training would make sense at this level of technology though.
Building the ship
After the pilot has finished the training and gotten good enough scores to merit a ship and has adjusted to being in the simulator cocoon all the time, the ships body will be assembled around the cocoon and the ship systems will replace their simulated replacements one by one.
The ship components would be built in parallel with the final step being fairly fast assembly from pre-fabbed components. The actual speed would depend on the maturity of technology.
Faster is better for business (the cradle for ship assembly must be very expensive and limited resource) but failing the assembly would probably be even more expensive so if the technology is new their will be an iterative process of testing and adjustment needed for every component and components will be added one by one.
With more mature technology increasing portion of testing and adjustment can be done before assembly and component can added in parallel allowing very fast construction.
I'd go from few days to few weeks. This seems very fast but most of the work would be done in parallel with the years of pilot training. You can even instead see the assembly around the pilot as a process of inserting the cocoon and the pilot inside it into the ship and get the time down to hours but nothing in the question suggests a need for such rush jobs and it would not fit the flavor of the setting.
But this destroys the biological flavor!
Not really. We associate such things with mechanical or hardware solutions but they are actually technology neutral results of the economics. We just do not have enough control of the biological processes to do large scale construction with biotech. The people described in the question do.
With premises in the question a biological growing of a ship would not be a slow and awkward process, instead it would be faster, more efficient, and more convenient than our processes for constructing aircraft or buildings from prefabbed components.