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Surely if you have bullets left in the clip you can shoot anyway regardless of whether it’s started to reload?

That’s the whole point of automatic reload.

I had a similar problem a while back and the problem turned out to be in my mouse.

press F5

Quoted from Lithium

[…]

Yep Im having the same idea… And for others, the auto reloading prevents me from shooting, in a random manner… I spray my bullet then it stops to reload and block my shoot.

Its most likely your mouse not registering the clicks. If you have autoreload on, whenever the mouse stops taking in your clicks the gun will reload, giving you the illusion of forcing random reloads and not being able to shoot while reloading.
If not that, I have no idea.

Quoted from Lithium

[…]

I think you’re right. I can’t see any solution so I’m gonna replace my mouse with a new one today. I’ll let you know if it’s better then. And it should be cause I attend the LAN Gamers Assembly tomorrow :x

You might want to research the problem some more before you go out and buy a completely new mouse.
If this is only happening with your pistol or only happens in TF2 and not elsewhere, your mouse bugging is most likely not the source of the problem, but rather some tf2 setting that’s totally beyond me. You should try using your mouse on another computer, try using another mouse on your computer and try other USB ports.
Most companies are also very generous when it comes to faulty products. More often than not if you contact their support, go through the try and error process with them they will offer you to send them your device(with free shipping) and a replacement product once they receive it(in my experience that is).