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Regarding the Many World Multiverse, everything I've been reading (and searched on SE) seems to assume that each (parallel) universe is, at this moment, the same age.
For my story, I need this to not be true for my time-traveler. By making it not true, is it no longer within the scope of the Level III: Many World (Multiverse) interpretation of quantum mechanics? Is it within the scope of some other theory?
I don't think there's any reason you couldn't stagger "now" between different worlds, but you may have some explaining to do.
In most multiverse theories, the argument is that anything that could have happened did happen, just in an alternate universe. This includes all sorts of things that could have happened, and the vast majority of timelines look nothing like ours! Consider that there's been billions of years of events happening from the big bang to the present. That's billions of events that could shape the evolution of the human race... or prevent it from occurring at all. There's no way to know the numbers for real, but it's reasonable to assume that the number of universes where humans even exist are vanishingly small compared to the number of universes out there.
This is all fine and dandy for exploring quantum mechanics, but for telling a story it's inconvenient. You don't have thousands of pages to document all the failed and boring timeline jumps that occurred before you found somewhere worth going to. So typically worldbuilders creating a multiverse handwave in an assumption that there's some sense of "nearness." Each universe has some sort of "neighborhood" of similar universes but with some small change. Typically these something that's easy for the reader/viewer to understand such as "Germany won WWII," even though quantum mechanically it's not clear why that would qualify as "near" our universe. But, nontheless we use this nearness or sameness concept to make sense of our multiverse. It brings some sanity to the worlds as we explore them.
You will have to use this process in your own literature to tame the wilds of the multiverse. However, you don't have to use anyone else's definition of nearness. While it is common to assume "two points at the same 'now' are near to each other in the multiverse," that's really a matter of convenience, not a physical law. If you want to stagger "now's," go for it!
The one thing to be aware of when doing this is Sanderson's First Law of Magic, which I quote so often I really need to have some notepad file up with it rather than looking it up every time!
Sanderson’s First Law of Magics: An author’s ability to solve conflict with magic is DIRECTLY PROPORTIONAL to how well the reader understands said magic.
You will certainly be resolving conflict with your multiverse theories and whatever approach you develop to jump between universes. The better a reader understands this process, the more you can do with it. In your case, one question to ask yourself is "why isn't it different?" If timeline A is 5 hours behind timeline B, why is it that way? Why is it not timeline A 3 hours ahead of timeline B? Surely that is an equally probable staggering of "now!" If everyone is at the same time, justifying it is easy. If universes are staggered, you have a bit more justification to do. It's certainly not impossible to do, just something to be aware of as you go forward with a slightly less standard approach to multiverse theory.
There's even one which is contained within the scope of the one who seem to wish to use: Level II of Tegmarks hierarchy. Which states that within the multiverse, there exists a infinitude of universes having each different properties to our own. Any number of properties could explain why the entire causal chain of a specific universe would be out of sync relative to ours yet be effectively identical other than in that one measure.
If you don't want your "time-traveler" to be able to jump into worlds where the laws of nature are completely different (and where he would immedietely die if he were to go to them) and yet want an explanation for why he can't(or doesn't accidentaly do so), simply handwave the problem away by making whatever method you're using to hop realities only work with universes which are "compatible", that is to say ones with universal constants that are sufficiently similar to the original reality. This seems perfectly reasonable to me.
Short answer:
There are 2 cases: either the universe splits with each measurement, or the 'path of measurements' of each universe is already mapped out, even though it measurement outcomes still appear probabilistic.
In the first case, fundamental constants can be changed, allowing for different universes to appear to be different ages, although this is really a result of changing the (non-fundamental) laws of physics, so things just happen at different rates (if they even happen at all). Alternatively, handwave that the big bang is probabilistic and so the multiverse is constantly splitting into new universes, giving multiple copies of the same universes, only at different times.
In the second case, big bangs can happen at different times, so different universes can be different ages as required.
Long answer:
As details about the various interpretations about quantum mechanics are fairly complicated, let's assume that there was a big bang at a multiverse time $t = 0$ (ignoring all the complicated details that this brings up) and that time travelling from a universe with multiverse time $t$ in that universe will bring you into a different universe, also with multiverse time $t$ in that universe (if not, then you don't even have a problem to begin with). Now, there are two possible ways that things could go using the Many Worlds Interpretation (MWI):
Case 2 is straightforward - other universes can be younger or older than our own, depending on when it was created relative to our own.
Case 1 is probably what most people think of when they think of the MWI. There are a couple of ways to solve this:
Things start to get a bit more complicated here, due to time. Here, a universe that only recently split off from our own is going to be almost the same as our own, including having the same time. However, a universe that split off right at the beginning may have different physical constants (actually a level II multiverse and entirely equivalent to level III). Now, if (as an example) you change the speed of light, $c$, things will start travelling through time at a different rate than in our own universe due to relativity, so although the alternate universe will have the same cosmological time, it will appear younger or older, although at the risk of the universe being considerably different to our own. Having said that, the universe is billions of years old and assuming you only want to travel a few hundred years back (or forward), the changes might not need to be that great. Although you're still messing with the fundamental constants of nature, which is messy...
What was the big bang? Well, no-one knows, except that it was extremely hot and that it suddenly expanded. Now, things like entropy would cause this expansion, which suddenly makes the whole process probabilistic. That is, at the time of the 'big bang', the universe splits into a universe where the big bang has occurred and another where the big bang hasn't yet occurred, Despite this, the 'multiverse time' is still somehow increasing (defining time this close to the big bang is an issue) and so within the multiverse, there are an infinite number of big bangs and so there is an infinite number of worlds just like our own, only at a different time according to that universe, even though the multiverse time is the same.
Piers Anthony's novel Orn in one of his early trilogies had a multiverse set-up where different parallel universes existed at different time rates. If you have time travel, there's no reason why there shouldn't be time differences between parallel universes.
In fact, if you were constructing a model to explain time travel a multiverse where parallel universes were at different times (or ages) would work well. Time travel in parallel universes has less problems than most other forms of time travel. No causality violations for starters.
Worrying about whether this concept works in or is contrary to quantum mechanics or any version of a scientific theory about multiverses is irrelevant. Unless you want to publish it as a paper in Physical Review.
If you are building a fictional world, you are allowed to make it work however you want it to; though it does make sense to try and avoid any obvious errors. Time relations or differences between parallel universes is an open question. Science doesn't have an opinion either way or another, therefore, you're free to do whatever you like to make your story work.
What is the “same moment” between different branches? If travel is possible between different points in time across branches, does the idea of synchronization even make sense? What kind of observer is saying the same time on different lines are “lined up” or “not lined up”?
In The Proteus Operation, connection between the two ends of an established time link (which will always connect different timelines!) was complicated by the fact that the past end was experiencing time faster than the future end. This would provide a sense in which different timelines share a present; the point at which the rates are the same at each end of the connection.
But the idea of a universal present moment doesn’t even exist in our normal single universe! It makes sense that there is no implicit lining up of timelines, unless you provide reference for what that actually means. And then you can push that meaning in interesting directions to make more surprising consequences or unusual situations.
One reason parallel universes are generally "the same age" is that it is often portrayed as similar universes being grouped closely together - variations on a single timeline being "easiest" to reach (usually demarcated by different decisions at critical points) - though this is generally for story-purposes rather than physics, as I understand it (it being more fun to tell stories of really-nearly similar universes, or really, really different ones).
That being said, I have read stories with different universes in different places in their timelines. Sometimes it's a function of events being placed differently - some event happening "soon" or "late" means the timing of effects and aftereffects will be shifted up or down the timeline, rushing or delaying the historical events that shape the setting. Or events, history, can happen slightly closer together, or further apart, so even though the broad strokes look the same, the "now" is placed in somewhat different places because the events are a bit shifted about in time.
Sometimes, it's a function of some really subtle alteration of universal constants - like each second being a quarter or tenth or a nanosecond longer or shorter, which may not make an observable difference in a person's observations, or life, or perhaps even a hundred lives, but over the universe's lifetime will lead to very similarly-developing timelines being in different places in their timelines (or even different ones, though usually you don't need to be meddling in universal constants to just make the universes very different). So the timeline can be off by a few years, or a few hundred, very easily - as long as the parallelism of the timelines being developed justifies the universes being "close together" despite the very fundamental differences in their universal constants.
Or to take it in a completely different direction, one of the theories of the multiverse have new universes branching off at critical decision points - thus the "new" universes are much younger than the ones they branch off of, even if they share the same long history up till that point. In case I misunderstood what you meant when you wanted the universes to be different ages.
If you want to head into the fantasy side of things, the Fae worlds are generally considered to be running at different time rates.
I see nothing wrong with different universes having different rates of time. None of them would know unless someone were to travel from one to the other and back.
To have any kind of "parallel" universe, the difference in time rate would be very small. After all being 1000 years ahead or behind is a very small fraction of 13B years. The time rate differences would probably not be noticeable (or even measurable) over short periods.
If the time rate difference was something noticeable (like 2x), you would have a case where the Sun had long ago gone nova or, in the other direction, the Sun may be condensing our of interstellar gas.
