As written, yes—but you might have to steal it. Giving an item to someone is a move action, even just dropping an item is a free action, and it says the only action the copy can take is aid another. So you’re left with things like stealing or disarming to try to get a hold of anything it’s carrying.
The other side of the coin is that you have no control over how the ally across time’s items are actually arrayed—just because you have, say, a useful wand in-hand at the moment, doesn’t mean the ally does. In its timeline, that wand may not have been useful at the moment. The wand might be packed away, where stealing it is going to be very difficult. That kind of thing is left up to the GM, which makes it very hard to abuse it.
It also could make it difficult to use ally across time for its intended purpose, since your ally cannot draw a weapon and might not have had a weapon in-hand just as easily as it might not have a wand. And, if you did get your hands on some powerful item—something more powerful than a wand, probably—success in this endeavor could well be game-breaking. I mean, if nothing else, if you came across a scroll of wish, you could easily cast ally across time in some safe space to try to steal a copy of it and try your hand at the UMD or CL check over and over. Anything that can result in endless wishes should almost-certainly be banned.
So between the fact that, strictly speaking, the GM can mess with the spell in ways that make its intended usage pointless, and the fact that, if they don’t when you’re trying to do something like this, you could break the game, suggests to me that we’re all better off if we just agree to not. The houserule I’d suggest here is, roughly, as follows:
You can, within reason, dictate how your ally across time will actually be equipped among the items you have on you, so that it can offer the best aid another possible or whatever. But in return, the ally across time’s items are simply off limits, except insofar as it can use them to aid another or use a teamwork feat or whatever.