It looks like the focus of this question is not on the creation of the computer itself, but on how to reliably provide power for it. So, I'll answer that.
As alluded to in a comment on another answer, an array of thermoelectric modules would be a great solution for providing a nice clean supply of electricity, and would run off pretty much anything you could burn to create a sufficient temperature differential. However, they're made of exotic semiconductors. Now, if you are also constructing the computer, then you can build these easily, or the same kinds of doped semiconductors that photovoltaic cells are made of, and then you're done -- but that's boring! What if your nanites could only do more simple construction? The problem is two-fold: power generation & power storage.
For power generation, the main things you’ll need are copper wire for the stator windings and permanent magnets for the rotor. Getting a sufficient quantity of refined copper to work into wire would definitely be a challenge (see the later part of my answer for a possible solution), but my gut tells me that permanent magnets of sufficient size and quality might prove to be your biggest challenge; you could use perhaps try have your nanites refine naturally-occurring lodestone. * applies handwavium * Presuming that you could manufacture a simple electric generator, the problem then becomes one of turning the rotor.
You've got several options for this; obvious choices would be a water wheel or a windmill, depending on the topography & weather of your location. Strictly speaking, the Bronze Age began around 3000 BCE and the earliest examples of this technology being practically applied were the 3rd century BCE and the 9th century AD respectively, so you'd definitely need to bootstrap some local advances in woodworking & stonemasonry, but your time traveller has obviously brought with them other advanced engineering concepts, so these would not be incongruous.
Other than that, a clever low-tech method would be harnessing gravity. A heavy weight, winched to the top of a tower, could pull a line connected to simple wooden gearing, ultimately turning the generator's rotor -- a rough, large-scale version of a weight-driven clock mechanism. Another possibility would be constructing a Stirling engine, but I suspect that even a crude one would require things like sheet metal or basic machine tools.
Providing a steady, reliable flow of power is also going to be important, so let’s look at building some batteries. Voltaic piles are quite simple in terms of battery chemistry -- alternating discs of copper and zinc, separated by cloth soaked in salt water. However, they're not rechargeable! So, let's look the oldest and most venerable rechargeable battery chemistry in the book: the lead-acid battery.
Sulfuric acid was available to medieval chemists, and is pretty straightforward to make -- a later 17th century process was simply burning sulfur with saltpeter in the presence of steam. No need for handwavium here! As for the lead plates, you could set your nanites to work precipitating the metal from water. It might take a while, though; according to one source, lead is only 2-30 parts per trillion in seawater. Freshwater is an order of magnitude better, 3-30 parts per billion. A gram of lead in a liter of water is a billion PPB -- so at an average of 15 ppb, you'd have to go through 67,000 L of freshwater to get 1 gram of lead. That's 2,366 cubic feet of water, so be prepared to wait!
Power consumption of computers ranges pretty widely. ARM-based boards like the Raspberry Pi use ~2W; a home theater PC would use, say, 13W; a laptop, 2x-4x that depending on the chipsets, and a desktop PC, potentially double again that of a laptop. Furthermore, power consumption under load vs. when the computer is idle can make a huge difference. So, when your power system is built, pick your computer hardware carefully!
