How long will the Universe's hydrogen reserves last for?

Tis, a good question. Two related questions arise from it. The first one is, will the hydrogen all be used up in finite time? The second related one is, will star formation completely stop in finite time? They sound related, but the first result doesn't necessarily imply the same result for the second, or vice versa. I.E. a low but nonzero gas density might possibly not allow further star formation, and maybe we could have no hydrogen, but have other types of gas (or even solid objects) still collect into stellar mass objects.

I don't know for sure the answers. The rate of star formation (and hydrogen consumption) could decline slowly enough as to never formally reach zero. Or not.

We do know that a lot of gas gets blown out of galaxies by massive stars, supernova, and black hole activity, and becomes intergalactic gas -usually staying within the galaxy cluster. On a long time scale this should eventually fall back into the cluster's galaxies. So I would think the star formation rate would have a very long tail.

The vast majority of hydrogen in the universe is in hot gas in galaxy clusters, or cold, extremely diffuse atomic hydrogen in the intergalactic medium (Lyman alpha absorbers). Neither population is likely to ever form stars, so I think the safe answer is "forever."