It's going to be slow, expensive and unreliable if it works at all. While in theory the computers could hide some of the problems, remember that computers are built, maintained and installed by people. So is software. If you don't let the people talk to each other, the system will not work very well.
The key problem is that no-one must be able to guess the nature of the system from knowing every detail of one part of it. That means the parts have to be fragmented to a high degree (even the parts that do the integration). So there will be a lot of communication problems, compounded by the need to carefully filter the communication to prevent anyone knowing too much. You can't have the "spy on webcam" people saying to the "store files" people "this is webcam video", even accidentally. That means every communication between developers has to be filtered by a human being. Which makes communication slow as well as uninformative.
Debugging will also be a nightmare since no developer can be allowed to see the system running, and each part will have to have its own debugging infrastructure. So you, developer of the "pipe an audio stream from the network to the user interface" component, will have to debug the "sometimes it stutters" bug without knowing what the user interface is or where the audio comes from (ie, what device is on each end). It's going to be very tempting for you to say "my log is clean, it must be someone else at fault". Everyone else will be saying the same thing.
Another problem is that the "inside" team gets very large, and the bigger it is the more management it needs, which makes it bigger. It may end up being bigger than the team that could have developed the whole system inhouse.
I've worked in a large-ish bank that had similar problems. No so much because they were a secret organisation, but just because they were big and some people had to deal with a lot of systems. So if a foreign currency transfer failed, some poor user was looking at a bank internal monitoring program that said "ERROR: transfer failed", when the international currency transfer program says "error: success" and the recipient bank is saying "we got your request, where's the money"... and so on. Their solution was to have a team of "insiders", like me, who were contractors hired to provide tier three helpdesk support, but from the inside. So we had admin rights to pretty much everything (just like Snowden and Manning did), and our job was to sort out things that the outside contractors couldn't fix.
I suspect inevitably a system like this will fail because there are huge incentives for everyone to "just make it work", and very few for "don't tell people the critical information they need to make it work". Spy novels fix that by using threats to kill and appeals to patriotism as incentives not to talk. But in the real world the people paying the bills (government) want results now, and "but there might be a leak later" is much less important than whatever silly event has everyone excited right now. Even when the event was Manning leaking stuff, the response was to give Snowden more access rather than less. Ooops.