Travel by train to Chamonix is not a viable option if time is of the essence.
Rail travel is relatively slow because there is no direct route. The Mont Blanc Tunnel is a road tunnel with no rail option. The rail route is up the Domodossola Valley and through the Simplon Tunnel into Switzerland.
From Torino Porta Nuova take the train to Navarro and change for Domodossola. This is the fastest and cheapest routing. Alternatives via Rho-Fiera Milano or Milano Porta Garibaldi are slower and more expensive.
From Domodossola the route passes through the Simplon tunnel into Switzerland. Typically, a change is required at Brig with onward travel to Martigny and a further change of trains onto the Mont Blanc Express.
Check TMRSA, Trenitalia and SSB websites.
