How long would a layman take to understand mathematically the standard model?

There's no quick shortcut to learning unfortunately but if you're really interested and motivated the best way to go is to check some uni curricula in physics and follow them by yourself. It will probably be calculus first (you can study it on Khan academy online) then classical mechanics (the boring Newtonian way), thermodynamics, classical mechanics (the Hamilton/lagrange way), electromagnetism, more advanced topics in math like topology, Hilbert spaces,..., quantum mechanics, classical field theory and statistical mechanics (I'm following my curriculum which is quite complete for a first cycle IMHO). You can study on books which are recommended by the Unis and try to follow them

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