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.
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):
- Every time that a measurement (in the quantum sense of measurement), the existing universe splits with everything in the new universes being the same, except for the measurement outcome, so that everything that could happen does happen in some universe within the multiverse, which originated with the big bang
- There was a big bang for each universe. The measurement outcomes for everything that will be measured in each universe is already 'determined' in some sense, even though it appears to be probabilistic (a bit like the two state vector formalism of quantum mechanics). Perhaps each black hole contains another universe or somesuch, with the multiverse being the originally created universe, containing all the other universes
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...
- (much neater, although currently even less provable science and requires more handwavium.)
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.