Judging only by the information you've given us:
Their speed and maneuverability are both very high. If a military had the ability to produce these kinds of vehicles in different sizes, with different levels of firepower, what purpose would any other sort of vehicle have?
It depends. Your flying disks seem to be very fast and very agile, but that doesn't cover up all the desirable properties of a military vehicle.
Different designs come with different properties. You mentioned speed and maneuverability, but ask yourself:
What about resistance? How resilient are your UFOs? Maybe the disk design, at best of your factory production capabilities, doesn't allow armor plating. Maybe disks can't be thickly armored without losing said speed. Maybe UFOs are inherently more fragile, since you have to spread your armoring equally on the whole circumference.
What about cost? UFOs may be sensibly pricier in comparison with other old, more well known and optimized vehicles. Most armies will still want to have UFOs around, but if they aren't cost-effective, you don't want too much of them. You could have, for example, a battalion of multi-purpose top of the line UFOs and regular scores of warplanes to throw at the enemy.
What's the status of anti-UFOs technologies? We're talking armies, so we're talking war. So it's a matter of attack vs defense. If the only way to fight an UFO is with another UFO, everyone will want them. But is that likely? Look into anti-aircraft technologies. If someone can cover a base with possibly cheaper stationary anti-UFO cannons to make it UFO-impregnable, then you may want to think about something else.
What's the status of UFO-compatible technologies? Similar to the one above. Are you sure that all the modern, best weapons/shields/engines/systems can be mounted on UFOs? Maybe you can't mount, let's say, missile launchers on UFOs. Maybe you can, but in a slightly less powerful version. Maybe you can't make UFO stealthy and so on. What fuel do they use? Again, is it cost-effective?
The overall question you should answer is, finally, are they really multipurpose?
You mentioned that you are able to produce them in various sizes and different level of firepower, but you have to draw a line and decide what they can and they can't do (if they are absurdly cheap and we can scale them down to hand-palm size, I'm considering having an UFO-drone to cut my beard with tiny lasers, instead of the razor).
UFOs may be really adaptable, but as a general rule, something built to be good at one simple task will be better at that task than one "very adaptable" vehicle. Tanks may still be better in city combat, submarines may still be better underwater, planes may still be better at specialized tasks (like traveling in high atmosphere or stealthy), other kind of spacecraft may still be used, and so on.
On a lighter side, if they are so common in your enviroment, should they be called IFOs (Identified Fliying Objects)?
After the Edit:
I'm adding this section since you edited the original question. Luckily for me, I think that most of my answers are still pretty valid. I'd like to add a point, since you said:
The military that possesses this technology is the only one in the world that has it.
Let's say that only military A as the UFO tech, and all the other militaries have different, comparable, but possibly lower vehicles. Then it's in the best interest of military A to keep the UFO as secret as possible. On the other hand, it's in the best interest of all the other armies to get their hands over the UFO tech.
This is simply a matter of warfare, were the unknown can easily wipe you out. There will be a strong push to get as much information as possible about UFOs, including proper construction plans.
This can be accomplished through spy work, or simply capturing an existing UFO.
Think about the WWII or the Cold War and how much stress was put into "not letting anything fall into enemy hands". The UFO may be complex stuff, but given the right data other scientist may replicate it.
So, while army's A may want to show off its powerful UFOs to intimidate potential enemies, it's best if they keep a close guard on them. In this scenario, employing lesser vehicles for menial tasks would be better: e.g., you don't send an UFO to do basic recognition, you send a plane.
Remember that the other armies just need a small UFO's going AWOL to start studying it.