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If the number of real cases double every N days, the number of people needing hospitalization is also likely to double every N days. If not, it probably means that the infection rate spread at different rates in high risk vs low risk part of the population, but unless those populations are strongly segregated from each other, that is unlikely.

This could go in both directions. In both cases, though it would require that the high-risk subpopulation is primarily infected by each other (or staff) instead of being infected from the general population. If high-risk individuals are primarily infected by low-risk individuals, the doubling time for each group should be the same.

Note that such an effect could go in both directions. The high risk subpopulation could get it at a higher rate the rest if we let the infection spread like wildfire in hospitals and elderly housing centers, or it could spread at a lower rate if the risk patients are kept isolated.