"Educated" Guess?
Okay, looks like an interesting game, so let's play...
On my monitor, your photo shows the projectile deformed & on a tape measure. Looks like it should straighten out to about 1 1/16 to 1 1/8 inch in length, as it appears to be an spitzer profiled soft point. Thats 1.0625" to 1.125", or a median length of 1.09375". This displays on my screen as approx. 4 1/2 inches. The diameter shows to be about 1 1/2 inches on my screen, which proportionally equates to about one third (4.50 divided by 1.50) of the length.
We assume approximations only, as you haven't measured the actual diameter or weighed the residual mass, it stands to reason that one third of your length will result in the diameter. Taking the middle ground, we guessed 1.09375 as the length, divide by 3, and end up with a diameter of 0.3645.
Applying this information to common bore diameters, we round-out to 0.366" (9.3mm) or so. Your bullet could actually be considerably different, as the photo image can be out of proportion due to angle, etc.
All of this is nothing more than an entertaining exercise without more precise information. If you care to get a more accurate measure of diameter and weight, you could at least establish a list of possible commercially loaded cartridges this bullet may have come from.
Now that I've come up with all this theory, I find the very reasonable & logical/scientific post by Michaelmcgo, above. I yield to his superior skill and resources, and I most heartily agree. Here in the United States, the popularity of the .338 is far more likely.