There are no rails anywhere close to Nome - indeed Alaska is not connected to the North American rail network. And I'm not sure the position is much better on the Russian side - it's a long way to Yakutsk (~2,000 miles). So it's a bit more complicated than 'merely' building a 65-mile tunnel (though that would surely be the most technically demanding part).
I think the initial idea would be to transport by road up to the railheads on each side. I wouldn't count on being able to take the train from St Pancras to Grand Central for a little while yet!
The anorak, please ...