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Exactly how long you can afford to wait for the Institution to spread from that one province will always depend on your current situation, so there is no one "correct" answer here. Also be aware that the current development of a province affects how much institution progress each dev click gives - the lowest-dev province or current cheapest per point won't necessarily be the best province overall, as you'll need more total dev clicks that might add up to more overall mana spent. The total amount of mana you need to spend to force Institution spawn in one province, before cost modifiers, is around 1900-2400 depending on starting development (the best starting value is actually 16 dev), but you can get that number down significantly from stacking dev cost modifiers.
If on the other hand you mean being the one in whom the Institution actually originates, that's another issue entirely. Each Institution has its own specific requirements for where it can spawn, which are flatly impossible to meet for some nations (Renaissance can only spawn in the Italy region, Printing Press only in the North or South Germany regions or any Protestant or Reformed province). Check the wiki or the Institutions menu in-game for more detailed information for each Institution's spawn requirements.