IMO most British Conservatives would be Republicans if they'd launched a US political career, and most US Democrats would either be Labour or Lib Dem in the UK. I'm sure there are exceptions but in general I think Democrats aren't motivated by the same sorts of things as Tories and vice versa, except of course where they're mainly motivated by whatever suits their own careers best.
I agree with you. I'd argue it would be far easier for a British politician to gain mild success in America than vice versa. The sort of US politicians who could climb their way the British ladder are the low-key but relatively high-quality moderates - .e.g. Greg Walden for the Republicans, and quite a few Democrats - someone like Denny Heck or Mike Quigley would fit in well as an average Labour MP. I vehemently agree. That a Labourite MP would join the Democratic party makes no doubt, but it doesn't follow that a Democrat would join the Labour party.
In fact, as far as taxes, business and interventionism goes, I think the vast majority of the house Dems would part ways with Corbyn and feel closer to a Cameron or a May.
Corbyn? Maybe
The Labour party at large? Not really.
I imagine a majority of Democrats would join the British Labour party, though there would be a very big minority that joins the Lib Dems of course (and a handful who join the Greens maybe?).
But I can see barely any joining the Conservatives.
Then again comparing politics between countries is hard to do.