By black hole most of us mean the Schwarzschild metric, but the Schwarzschild metric is time independant i.e. it represents a black hole that has existed for ever and will continue to exist for ever. So it has no age in any useful sense of the word.
The problem with real black holes is that, as discussed at some length following Hawking's recent paper, Scwartzschild observers (i.e. you and I) will never see a true event horizon form. The best we will ever see is an apparent horizon. So if you consider a black hole to exist only if it has a true horizon then we can't assign an age to a black hole because none exist.
I'd probably go along with the pragmatic view expressed in BMS's answer. If you take the black hole at the centre of our galaxy it seems a bit pedantic to point out that it only possesses an apparent horizon, and therefore isn't really a black hole, when its worldline certainly leads to a black hole whether we'll be around to see it or not. Most of us would judge the age of the black hole to be about the age of the Milky Way.
Viewed this way, you're really asking about the age of the black hole's surroundings, whether it's a galaxy like the Milky Way or a supernova remnant as in BMS's example. You'd judge this age as you would for any similar cosmological body.
Under General Relativity, Lorentzian wormholes (the kind that are traversable) require exotic matter (a kind of unobtainium which is not known to exist).
This is not true. A maximally extended Kerr black hole solution for instance has a traversable wormhole to a (different) external universe and doesn't require exotic matter. However you do have to traverse a region with a visible singularity and with visible regions where time travel is possible (where closed time like curves exist). It is unreasonable because it is an eternal solution so the black hole never formed from infalling matter.
On the other hand, we know black holes exists and these form from the collapse of large stars.
External observers, by definition, never see them form.
One of the differences that we usually associate with wormholes is that they do not have an event horizon, and are disallowed by topological censorship theorems.
Most topological censorship is based on changes in topology. And there are traversable wormholes with event horizons, but that is usually interpreted meaning they go to different universes and the alternative seems to be closed time like curves that go through the wormhole. And the latter happens anyway, just if they are different universes the closed curves loop around only in the inside regions and never in the outside regions.
But what about black wormholes? is this concept a meaningful one? what I'm thinking is a Lorentzian wormhole that is unidirectional, basically you fall in an event horizon, but instead of finding a singularity, you enter a throat and exit on the other side.
This seems lime a novel (possibly nonexistent) solution if you are trying to avoid using exotic matter.
From the exit mouth, you can 'see' the other side, you can even traverse the wormhole back to the entrance side, but you are unable to send anything to null infinity because of the event horizon.
This is very confusing. If you can cross a surface going in both directions it doesn't sound like an event horizon.
Would such exit side be essentially the same thing as a white hole?
White holes can have there own type of event horizon, ones where you can cross from the inside to the outside but not vice versa. This is the usual type of horizon to cross into an external universe, such as in the maximal Kerr solution.
Best Answer
Personally I'm not entirely sure how much technical detail you require, only in a generally digestible fashion. However, given that I have recently read answers provided to numerous questions regarding black holes*, formulated for the 'general reader' but by someone who, to drastically understate, certainly knows much, much more than myself, I'll quote:
This may be far less technical than you desire - let us know if so. Though, even so it is an easy read with enough of an idea and general knowledge to be contributed, I believe.
*Source: BBC Focus, Brian Cox, April 2011