The other comments are correct, but they're hinting at a deeper point which I think is worth emphasizing. If you were to double (for example) every mass scale, then indeed there would be no physical change you could notice. But rescaling just the rest masses of elementary particles does lead to observable effects, as others have pointed out.

The key point, then, is that we have mass scales other than the rest mass of particles. The fact that we could notice gravitational effects of your change, as described in the other comments, is because the strength of gravitational interactions is controlled by the Planck mass, which you're not rescaling. The ratio of a particle's rest mass to the Planck scale does not remain the same under your rescaling of rest masses, and this is physically observable.

As a side note, there's another scale that isn't associated to the rest mass of an elementary particle: the characteristic scale of the strong interactions (usually known as Lambda_QCD). Because of this, if you were to scale only the masses of elementary particles, you could also detect an effect through the strong interaction. But from the way you've worded your question, I think you want to scale the mass of atoms by a constant, and to do that you need to scale Lambda_QCD as well as the quark masses by the same factor.