When we look at particles at CERN and similar places, we are only looking at the present values of these "constants", of course.
One could imagine running experiments over so many years that any change would be apparent, but we are hardly there yet. Cosmological times scales are long, human time scales are short.
Fortunately we have a sneaky shortcut.
We can look at stars and galaxies far away. When looking at something ten billion light years away, we are in effect looking at how physics worked ten billion years ago. And that was only a few billion years after the Big Bang. Surely any difference would be easy to see.
But what do we see? We see stars that behave just like the stars nearby, we see galaxies behaving just like our own.
Now, we don't see very much at those immense distances, but light spectra can tell you a lot about both how much there are of each element and how those elements are behaving.
Everything looks normal.
Moving closer to the Big Bang we have the hydrogen and helium percentage of the universe. It is assumed (for good theoretical reasons) that an equal number of protons and neutrons were created at the Big Bang. Then some neutrons decayed into protons. After a few minutes everything had cooled down enough that protons and neutrons could combine in helium. Unlike bare neutrons, helium is stable. Because of neutron decay there is a surplus of protons, which is also called hydrogen.
Physicists have calculated how much helium and hydrogen there should be based on the present values of the "constants", and the results match what we see around us.
As @kangermu wrote, cosmologists consider the possibility of a change in the "constants" for the very first fraction of a second after the Big Bang.
That is possible. We will probably never know for sure. Some people might find that unsettling, but what can one do? It is better to shrug and get on with life.
As for the future, these "constants" have been very close to constant for billions of years. We can expect them to hold for our life times.