What we're looking at is a computer architecture that is derived from the architecture of the brain. Since it is a computer architecture, it could take advantage of all kinds of optimizations and extensions that computer architecture does.
Given that, let's take a look at how real low-level computer networks work. In a modern automobile, for example, there are dozens of computers that have a specialized task. One might control the seatbelt light. If the seatbelt isn't buckled and there is someone sitting in the seat, illuminate the seatbelt light. May not sound like much, but consider that there are plenty of microcontrollers on the market meant for embedded applications that can be purchased for pennies to a few dollars in bulk, and what's the downside?
Other systems make use of a variety of processors that all work toward the same goal, sometimes with specialized purposes and other times just to achieve concurrency. One example is the common CPU/GPU setup. But looking more closely, both a GPU and a modern CPU are designed for concurrency via parallel processing. I'm no expert on the human brain but from what I do know, it seems like it functions--at least on some level--similarly. There is a basal ganglia which offloads commonly-repeated tasks from the conscious mind, hence the famous human ability to walk while chewing bubble gum. Even though consciously doing more than one thing at a time is often difficult, it becomes "mindless" (so to speak) to perform common tasks like walking, so these can be performed concurrently with other tasks.
My thought would be to give each body its own equivalent to a basal ganglia loaded with many routines--even some complex ones such as speaking--and have the conscious part of the brain control it remotely, giving it tasks in much the same manner that the human brains relegates tasks to its own basal ganglia.
Another problem would be input. At a certain level, we humans seem to have the ability to process input sub/unconsciously, such as when something is so ordinary that you don't even take notice. But this only proves the idea that our ability to focus on input is limited. This can be easily observed anyway by trying to read a book with too much background noise. This matter of unconsciously filtering out unimportant input can be relegated to the bodies too in order to offset the load to the conscious brain, but...
The problem with all of this is that this is what our brains do anyway. They already offset these tasks to the sub/unconscious, which means that they only deal with what requires conscious attention for one body. So even by doing all of the above, the bottleneck would still appear to be one body. Beyond that number, we would have the same problems we already have with multitasking, only now the problem is even greater since we have multiple bodies. Simple things like deciding where to walk next would have to wait in queue while we tell a different body where to walk next. We would need to handle each task in rotation. This is actually exactly the function of an Operating System.
I think it would be too much of a stretch to introduce parallel processing to the conscious mind, as that would bring into question whether or not the individual really has one mind or several (actually, this could be an interesting philosophical direction to take your story if you wanted), so the only real solutions are either to increase the threshold for consciousness or to increase the processing power of the brain.
In the first case, your drone bodies would handle much more on their own. This means things that would have otherwise grabbed your attention will now escape your detection and your basal ganglias will just autopilot through them, for better or worse. An increased incidence of "brainfarts" will occur.
In the second case, the already massive processing power required to simulate a human mind in a single thread has to be increased further, leading to astronomical clock rates, which raises a whole new set of problems with computer engineering, including cooling (fast clock rates cause lots of heat; see this article) and relativity (light itself only travels 1 foot per nanosecond; the physical size of these processors will have to be tiny).