Check out my other guides:

• BMP-3 guide,
  
• Vought F4U guide.

Long story short: the secret lies not only in fast reaction but also in optimizing your movements. You can skip to the next section if you're not interested in details.
Anyway:

1) do not use the "animation speed-up", or "maximum speed", or "skip" button - they fast forward the time along with the animations and thus don't help you (in fact, they make you spend more total time than you normally would, because you no longer have enough time to think during the animation, or rotate the camera, or at least move the mouse pointer over the next part),
NOTE: this doesn't apply to Super Game mode, where you have to accelerate each part manually,

2) minimize your camera movement. Moving the camera during the animation is okay, but if the animation is too short and you have to change the view angle after it, you will lose time. So try to move the camera only when it's absolutely necessary and preferably during long animations. You don't have to see the whole part right in front of you; you can try clicking the necessary parts through various apertures, peeking from other parts, etc. You only need to see a few pixels of a part in order to select it. Possible exceptions are models with hundreds of parts which create a mess on the screen or models with really small parts that have to be clicked very precisely,

3) research has shown that roughly 90% of gun models can be disassembled and then assembled with no more than a couple of camera movements (including the initial unloading sequence). Nearly all camera rotations are required for disassembly, while the assembly is almost always possible with one view angle, regardless of the model difficulty (with only a few exceptions). The table below contains a minimal number of view angles necessary to complete the Game (disassembly + assembly). According to the current statistics:
1 view angle is enough for 90 gun models,
2 view angles are required for 97 gun models,
3 view angles are required for 8 gun models,
only 2 gun models require more than 3 view angles (ZIS-3 and Flak),

4) mashing the left mouse button continuously can lead to losing Serenity and, probably, Sniper but you don't need those for Timed III, and it can surely shorten your total time if you memorize the order well (it will do you good to find out which parts are independent of each other and can be processed in any order and which cannot),

5) also, there's something I didn't realize until T-72 appeared: the timer doesn't start when the unloading sequence starts... when the loading screen appears, time is already ticking. Now, usually, the Game mode takes about 0.5 seconds to load (for T-72 it's about 20 seconds), so it doesn't seem much. But if Timed III is very strict (see below), even half a second of delay can cost you the achievement. So you could try hitting Esc and then selecting Restart as soon as the unloading sequence is finished, it will save you a little bit of time,

6) some guns have more strict requirements for Timed III than others - there are models which you can easily beat 2 minutes faster than Timed III, and there are those where you click like crazy only to be a couple of seconds faster. In my opinion, this isn't really fair, because you don't know beforehand how difficult Timed III is for a particular gun. However, I can help you with that information as well. The table below has a column named "Timed III". The number in it is obtained as (A-B)/(2*C), where A is the time required for Timed III, B is the ideal time that is shown in the demo mode, and C is the number of clicks required to disassemble or to assemble a gun (the number of clicks is present in the table as well). If you're scratching your head trying to figure out what it means... well, it actually means what delay (in seconds) you can afford before each click in order to get Timed III. For example, if the number is 0.5, it means you can think no longer than 0.5 seconds (on average) before each click. The following estimation is totally subjective and is based on several hundred hours of practice:
less than 0.4 - impossible (thankfully, there are no such models... yet),
0.4-0.44 - torturous,
0.45-0.49 - painful,
0.5-0.54 - severe,
0.55-0.59 - challenging,
0.6-0.69 - doable,
0.7-0.79 - moderate,
0.8-0.89 - simple,
0.9-0.99 - effortless,
1 and more - laughable.

Finally, here are some videos for you. Click on the model name to open its Youtube page. I'll appreciate any advice and recommendations.

Easy means less than 30 clicks.

Normal means 31-50 clicks.

Hard means 51-125 (maybe 120) clicks.

Very Hard probably means 126-200 clicks (exact number unknown, too few models available).
A minimal number of view angles hasn't been determined and isn't too practical.

Insane probably means more than 200 clicks (exact number unknown, too few models available).
A minimal number of view angles hasn't been determined and isn't too practical.