Noise comes into wiring in basically three ways:
1. Inductive coupling - an AC magnetic field near a wire makes a little transformer out of it. So running a wire past, say, a motor or power transformer of another device could induce a small AC voltage in that wire. What's important here is that you have to have the wire in question in a significant AC magnetic field. That doesn't happen around other power wires because there are two of them, and the fields around them mostly cancel each other. Another fact people don't seem to know is a shield around a wire is invisible to a magnetic field, and so is useless in this case.
2. Capacitive coupling - an electric field around another wire nearby creates a capacitor with the wire. What's important there is, capacitors pass high frequencies more easily than low ones, and 60Hz is a very low one. You could have problems around a switching power supply in a computer, but again, not so much around other wires. Shielding around wires does block capacitive coupling to the inner wires, but real capacitive coupling often isn't the big problem anyway.
3. RF pickup, the wire acts like an antenna - RF energy from big transmitters, like radio and TV stations nearby could be a big problem. Shielding will help keep the RF off the inner conductor, so long as the shield is well grounded, but the shield also becomes the antenna, so if it's not grounded well (from an RF standpoint) you've got a nice radio receiver there too. Same issue for local sources, but things like cordless phones and WiFi isn't usually an issue because they are well into the ultra-high frequency range which doesn't penetrate anything very well, and are very low power. Unless you run a wire right next to a device like this, it's probably not going to be an issue. Cell phones are different, and can be trouble, but the problem isn't the wire, it's the device receiving the signal.
So, wire shielding only helps keep capacitive coupling down, helps with RF if grounded, and not at all with inductive AC signals. What does help is that the device driving the wire has a low source impedance in the range of the interfering signal. So if you're using a preamp to drive a wire, and the preamp has a source Z of, say 100 ohms, it's pretty hard for an a signal entering the wire to have much effect, as it's looking at a 100 ohm load. If your preamp has a source impedance of 10K, that's a very tiny load, and an inducted signal won't be loaded much by it. The other important factor is that both sending and receiving devices share a solid common ground, which in a large home could be hard to do. Conduit isn't that great a ground, for example. If the receiving device is at a low ground potential, but the sending device is not grounded, it could be imposing a rather strong AC voltage on the wire. Shielded wire won't help here either.
The only way to really keep outside signals out of your wiring over long runs is to use balanced lines. That's where you have two signal wires, neither of which is grounded, and possibly an overall grounded shield. Since the wires are twisted together, they get the same interference. When they come to a balanced input, though, the input is looking only for differences between the wires, so anything common is ignored. As an example, the entire hard-wired phone system is balanced, unshielded twisted pairs over miles and miles. Mostly, no crosstalk. The key here is to use balanced lines, you have to have a balanced sending device, a balanced receiving device, and the correct wire. The OP doesn't have any of this.
So, it's not necessarily the shield or lack of it, it's the impedance of the circuit, grounding of the devices, and proximity to large interfering signals, and the input circuit of the receiving device.
With all that as variables, you pretty much can't predict if your unshielded wiring will work out. You just have to try it.
