Forgive me if this carries on a bit. I tried to be comprehensive and remain brief. Comprehensive won out in the end.
I'll define a nanomachine as an artificial mechanical construct to the scale of a nanometer (10−9 m). I would assume anything measured under 100 nm could be considered "to the scale" of a nanometer. I'll use a 10 nm3 block as a general size for one of these nanomachines.
This size brings problems with mobility, precision movement, available construction materials, and versatility among other issues. I hope to cover those somewhat thoroughly in the answer.
Also, there are a wide variety of potential uses for these machines, so for simplicities sake I will assume you are dissolving bad guys and their stuff with these little destructo bots or building yourself something awesome with them.
Let's hit the first topic: How Fast Can They Move?
Often nanobots are introduced into an already existing system (usually aqueous) and serve a specific purpose within that system. I think comparing them to catalysts, or cells is a pretty good way to look at it. With that being said, almost nothing of this size moves on it's own in nature. The only self-propelling elements at this level that readily come to mind are viruses and bacteria.
The fastest speed I can find for bacterial movement is 200 microns/second (a micron is 10−6 m). For something 10 nm in size that is an incredible rate, though not a lot of overall distance. If your machines were to move on their own I arbitrarily think up to doubling this could still be considered believable. I submit a max self-propelling speed of ~400 microns/second.
However, systems themselves can move much faster. For example, the highest blood flow rate in the human body reaches ~180 cm/s. If you were to transport nanomachines in a similar way (expelling an aqueous solution through pressured valves) you would not have to bother with aiding their movement. This leads to the maximum speed that a thin aqueous solution (like an alcohol) could move. However, a reaction between the nanomachine and whatever it is eating needs to happen. I would again apply my arbitrary double rule to the blood flow rate since countless reactions happen while blood is moving. System propelled speed of ~400 cm per second. That gets you magnitudes greater in nanomachine speed.
Really, depending on the activity the machines are serving this could vary greatly. In essence, if you propel them, they are limited on the speed of the action they perform. Seeing as some chemical reactions are basically instantaneous (photo decomposition) while other physical reactions can take millenea (forming diamond), your variety is huge. I'll address this later though.
Now we move on to the next topic: How fast can they chew through something?
This is also pretty variable. There are so many surfaces they can chew through, from weak fleshy membranes which even mild chemicals destroy instantly, to a slab of tungsten that could possibly survive hitting the surface of the sun (for an instant). I'll keep any calculations vague to accommodate this.
I'll settle on magnesium metal since it is somewhat similar to aluminum (one of the most commonly used metals in modern manufacturing) and I found data for it without needing a lot of research. If you react magnesium (also aluminum) with Hydrochloric acid (HCl) it quickly dissolves. I found this data from an experiment measuring the rate of dissolution for magnesium in HCl. Powdered samples dissolved within 30 seconds when submerged in 3 molar HCl. It is possible to get higher molar acid, with a max around 12M. It appears that even with the strongest acid the max speed of dissolution floats around 15 seconds. These samples could be equivalent to maybe 1cm^3 of solid metal. Because solid metal has less surface area I'll use my magic doubler and give a value of 1 cm of metal surface dissolved every 30 seconds. Assuming we would only use nanomachines if they were more efficient than this acid, I'll double efficacy, leaving 1cm of metal dissolved per 15 seconds.
Now for your last question: How fast could they build a macroscopic machine? (if I got that right)
This is where I think nanomachines are a bit over-hyped. There are a couple of issues that we run into.
- Where do the machines get the materials to build with?
- How do these machines transport these materials?
- How do you give detailed instructions to the machines?
With these questions in mind, I am giving a lazy answer: Don't use nanomachines for manufacturing
Right now we have so much available that is rapid, precise, and simple. Robots already pump out parts, devices, and machines at incredible rates. These parts are often on a macro scale, and when they are not, the process is still incredibly efficient. I mean, we have 3D printers that use lasers and gel to create anything that you can draft accurate to microns. It is fast and cheap and easy. I don't imagine that throwing nanomachines into the mix would make any aspect of manufacturing measurably better. The energy used to move materials with nano-machines and to give instructions to them seems to great to be of benefit.
Their use in creating drugs and chemicals is already pretty amazing. They can clean materials or products, treat materials for specific uses, and do a host of other things, but as far as building larger machines themselves goes, it does not seem too useful (please list any good uses you can think of in a comment).
Maybe a better idea would be like the nanobots found in Big Hero 6. Instead of building a macromachine, perhaps these nanomachines could combine their efforts and function as a macromachine collectively.
Altogether my answer comes to this:
- Self Propelled Speed: ~400 microns / second
- System Propelled Speed: ~400 cm / second
- Speed to eat metal: ~1cm / 15 seconds
- Speed to build a macromachine: Not useful to build macromachines
This is all very general, and depending on specifics could vary quite a bit. I hope that helps.