Construction times for urban settlements, urban investment, agricultural investment, and land improvement?

I take agricultural investment to be essentially money and resources that are handed to families directly or indirectly to settle in the domain, build their own huts, and purchase agricultural tools, so it takes effect immediately.

When it comes to founding and expanding urban settlements, urban investments (1,000 gp for 1d10 families), and land improvement (25k gp for +1 land value), I am unsure whether I should count these as taking effect immediately, or require construction times analogous to what we do with strongholds.

There is a case to be made for hand-waving the construction times of improvements and investments, that such large scale projects should not require the kind of management that goes into building a stronghold: The ruler just throws a bunch of gold at projects over a large area, effectively employing most of the local population, so construction time should not be that much of a factor.

On the other hand, the infrastructure for a metropolis “sprouting” off the ground from one month to the next does not seem to make much sense. Doesn’t investment for building a settlement largely go into town walls, moats, and fortifications?

But if time for construction is required, does this mean that people won’t be able to move into the new settlement until it is done?

I think I will house rule that money paid up front to build a settlement will enable the population to move in immediately, but that construction time still needs to be accounted for in case the settlement is besieged.

(As an aside, the amount of money that can be spent monthly on urban investment is capped at the domain income, but agricultural investment isn’t. Is this intentional or an omission?)