I'm in the UK, so I'm presuming I'm buying by mm thickness rather than gauge?
Not necessarily. AIUI, the UK have their own "wire gauge" standard, which is similar to, but not exactly like, AWG (American Wire Gauge). So you might encounter that as well. And depending on what sites you're searching, it's entirely possible that you'll still encounter wire denoted using AWG. Of course you can probably choose to explicitly search only by mm diameter if you want, but I'd guess casual searches will turn up a mixture of different denotations. Not a big deal though, as finding conversion charts is pretty easy.
Only a few, 5V, total load will be around 100mA. I want something not too chunky as there's going to be a lot of wire in close proximity, but thick enough to do the job.
If you know the current load you'll be dealing with, you can explicitly work out the required wire size. There are charts you can search up that show the current carrying capacity of various wire sizes. Or conversely, you can look up the resistance per unit of length for a given wire size and work it all out from there using Ohm's Law (assuming you know the expected voltage). The goal ultimately is to find the minimum required wire diameter / gauge.
Once you know that, you do have a few other choices to deal with, like insulation. But at 5V, any insulation is almost certainly going to be way more than sufficient. This starts to become more of an issue when you start messing with higher voltages and need to be very explicit that your wire is sufficiently insulated to deal with the voltage it's tasked to carry. Other than voltage rating, another factor to consider with insulation is its temperature resistance, and how supple it is. I doubt you're going to be pushing any temperature boundaries in this controller, so that's probably not a big issue. Just note the suggested max temp for your wire and double check that you don't be exposing the wire to anything above that.
Silicone insulation is usually a very good combination of suppleness, voltage rating, and temperature capacity (which is why it's so often used for high quality test leads) but it's usually more expensive than some of the cheaper plastic insulating materials.
Also: note that in the above paragraph(s), I glossed over any reference to wire
material. That's because, while various wire materials are available, I don't think there's any reason at all for you to consider anything other than copper wire for your application. But if for some crazy reason you did want to consider, say, aluminum wire (don't though), the charts I mentioned can be found where the wire material is one of the factors. But just use copper, really.
Stranded vs. solid core wire is another choice you'll need to make. Solid core usually has a little less resistance at a given size, and can sometimes be easier to connect to things without crimping or soldering on an explicit connector component of some sort. But stranded wire tends to be more supple and easier to work with. For an application like you're describing you could probably get by with either.