Is zakat due on capital invested in building a hospital or on its expected profits after a year or more, knowing that the money cannot be withdrawn before two years?

If you contributed to the project with the intention of benefiting from its yield, not for trading, then the share is subject to the zakat of productive assets. This means there is no zakat on the principal of the share itself; rather, zakat is due only on the income derived from it, once it reaches the nisab (minimum threshold) and a hawl (full lunar year) has passed over it. However, if the intention was to trade in the share, then zakat is obligatory on both the principal of the share and its profit (zakat of trade goods).