Going by train across the border is risky as they can remove you for the visa issuance and the train could leave without you
{SNIP]
I would take a bus into the US (megabus and NEON and cheaper and faster than the Amtrak BTW).
And you don't think the bus might also leave without you? Why do you think a bus would wait around for British passengers to process their paperwork but a train would not? Frankly I think the OP could use either bus or train though the train is bound to be about 100x more comfortable.