Could hydrogen replace oxygen in cellular respiration?
No, hydrogen could not replace oxygen because it has entirely different characteristics. The most important one is probably its electronegativity - oxygen 'pulls' electrons much 'stronger' than hydrogen.

Basics: Reduction potential

Oxygen is the so-called terminal electron acceptor of the electron transport chain in eukaryotes. You can see "reduction potential" as a kind of stored "energy" which molecules have, similar to the power stored in batteries (very similar actually). To make this text a bit shorter I'll call it "RP" from now on.

One maybe confusing detail is that a substance with low RP has "more energy" than a substance with high RP, so it is the opposite way of thinking.

In very generalised terms, metabolism means that molecules with a low RP (glucose) are oxidised (burned) and turn into molecules with much higher RP (CO2). Coupled with this, a different molecule with very high RP (oxygen) is reduced and becomes a molecule with slightly lower RP (H2O). (You may have heard this before - it's called a redox reaction.) *

The important part is that the RP "released" by the oxidisation (burning) is larger than the RP "taken up" by the reduction. The surplus leaves as energy - heat and light if you just burn the glucose. This is a spontaneous process, meaning it will occur just by itself - even if it takes a long time if nobody drops a match on it.

The idea of metabolism is to let that process happen - but use as much of the energy it releases as possible. This works by not just letting it burn, but intercepting that burning process at different stages so that at each step a bit of the RP can be taken off and stored in something else. This "something else" is NAD which I'm sure you've encountered before. Each step that glucose is burned down, another bit of NADH is made, which then has a respectable reduction potential.

NADH (leaving out NADPH here which is a bit different) is channeled into a process called oxidative phosphorylation which retrieves the reduction potential in an actual form of energy.

Basics: Terminal electron acceptor

Finally, the reduction potential I've been talking about all the time is really just electrons, involved in bonds which are "happy" to react. Passing down the reduction potential as I've explained is really a passing down of electrons into more and more stable, less reactive bonds. That's why it's called "electron transport chain". At the end of oxidative phosphorylation, those electrons are dropped onto O2 and make it into H2O. That's why O2 is called the "terminal electron acceptor".

Why Hydrogen can't replace Oxygen

Now to come back to why hydrogen cannot perform oxygen's function in our body. We use glucose as our source of reduction power and oxygen as our terminal electron acceptor. O has a high electronegativity (3.5) so it pulls electrons strongly towards it. H's electronegativity is only 2.1, so it's much weaker. O as a terminal electron acceptor works because it pulls them much stronger than H when they bond, so an O-H bond is almost like giving oxygen an electron. In order for hydrogen gas (H2) to perform the same function, it would need to be possible to drop electrons onto hydrogen in a bond where it pulls them much stronger than the other partner. They do exist, and such compounds are called hydrides. But the catch is: unlike H2O, these are normally strong reducing agents, meaning that hydrogen would rather not be in that bond. This not a feasible option for cell respiration, at least in humans, because it requires a lot of RP input. Making oxygen into H2O does not require a lot, it's a very cheap electron acceptor.

I hope I was able to put this in understandable terms. Let me know if I need to clarify anything.

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*It works with other molecules than glucose->CO2 / O2->H2O too, many prokaryotes do that and in fact that's how batteries work