A note on creature stock:
It might be more advantageous, if getting necessary creatures with all the requirements is rare, they may recruit creatures that have some of the ability with specialist tasks. Such as a handful of creatures doing crowd control, either against muggles, wizards or other sentient.
Past a certain level of intelligence, of course, you'd be dealing with sentience and all the ethical issues that come with it.
But wizards do use some creatures in this capacity, of course. Like security trolls.
I am assuming that if they are sentient, they volunteered or recruited the same way wizards (and witches but I'm not going to type that down every time) would be. They would also be compensated for their work, even if with not with money. No slaves, at least not ones that could be turned against the army with the prospect of freedom.
That last point could be tricky.
As much loyalty as the wizards themselves would have. Not necessarily absolute loyalty or absolute control.
Basically, I'm saying to avoid something like with what happened to the Dementors, who naturally allied to Dark Wizards over their regular wizard masters. Keeping them as prison guards might have worked but not when they started using them as security guards for school.
Or, for that matter, the endangered point if you're using magical creatures.
Not endangered to the point that their use and loss in battle would mean extinction for the species. Or if they can be bred (non-sentient), establish a breeding program that allows the species to survive even if all creatures in service die.
Also, I should have added: also not critical to existing magic infrastructure or security of the nation.
Rapid mobilization would be important and feasible for both forces. Arguably more important for the magical response one, as hostile wizards will likely have capabilities (like apparition) very well-suited to fast hit-and run tactics.
I should have added something like that for the magical response force, most wizards have brooms or similar vehicles. Flight or otherwise other ability to keep pace might or prevent escape would be necessary.
A squad of free, aggressive house elves with a tasers might qualify for example (I think). "Dobby says drop the wand and put your hands on your head or Dobby gonna shock ya!"
You'd need to come up with something that could stop small arms fire for the first force. Maybe some sort of magical material could function as a kevlar equivalent? Or a sufficiently strong shielding spell, of course.
I suspect that there may be already such creatures.
Or, if not, they could be
made impervious to bullets (and clubs and other weapons an angry mob would improvise) easily enough (potions that can be made or kept on-hand or at least in storage) and long enough to be useful. An hour's bulletproof-potion versus one that lasts days or even weeks.
Of course, the physical response may also be responding to things like rouge magic creature attack for whom a physical force is enough. So physical protection would extend to other things than muggles, such as claws or other physical weapons.
Not sure you'd want to use low-intelligence creatures for crowd control though. That sounds like a recipe for high collateral damage.
Hence the condition to have the ability to remain calm in the crowd or be kept calm, as well as be able to handle the crowd non-lethally. A dragon that would respond with talons and fire is a bad idea.
For the second force, you'd have a hard time coming up with something Imperius-proof, I suspect, although their are magical creatures which are spell resistant (including dragons, giants, and dementors).
The lore seems to have established that the killing curse is very strong. However, it seems Dementors are immune as they are not alive, so undead may have protection against it.
Of course, it is also sufficient if a creature can be made hard to hit in the first place, such as being invisible, able to instantly teleport or otherwise have measures to block. Or it may not even be necessary.
The point is that a more educated wizard would not be able to disable them with just one spell or two.
Keep in mind the numbers issue. Your "Muggle-specialist section" might be just a few guys.
I am not imaging this force large to begin with. I imagine a hundred or two combat-ready units in total, with a squad or two of enchanted muggle units.
In practice, I'd imagine this as a reserve force that has been gathered together in a Hodge-podge way. A bunch of Wizard War veterans browsing trough the wizarding world's miscellaneous and odd bits, gathering useful weirdos and others that could be hammered into an effective force after much training and effort.
The real question would be need. Who are you going to use the force against? Death Eaters? Small enclaves or magical creatures that had enough? Invading wizards? Are wizards likely to invade at all? Can they even invade? Is there an army of fairies that the UK wizarding world has reasons to worry about?
Because you tailor your army to the enemy. If you can't tell who the enemy is, you either need an intelligence agency first or ask whether you need an army at all.
The thing about the magic society is that everyone in it is qualified to do magic to a certain level. So a citizen militia makes more sense than it does in the muggle world. Aside the Death Eater, the Ministry seemed to have most things in hand.
Beyond that, I think the wizards have a pretty secure form of communication as is- the two-way mirrors I mentioned above (Sirius gives Harry one in book five, and it comes in handy in book seven). I don't recall their being any mention of any way to spy on a communication using that means remotely. Though if its just a two person thing, then you can't do conference calls.
I was thinking smartphones actually, that could replicate that functionality. If you could encrypt the calls and data sent trough, it would be fairly secure, especially against wizards. They could ask some help from MI6 and MI5 on that front. I don't know whether there is a company in the UK that sells secured phones.
Consider other things they could do: they could keep track where agents are, help them navigate, allow them to send and receive digital communications, etc.
Then their are a number of what might be termed civilian militias and local private security forces (the Centaurs, the Goblin Guards on Gringotts, the Order of the Phoenix, Dumbledore's Army, etc.). And their is a major terrorist organization (the Death Eaters and their lackies), which at full force could be considered an army.
