You've got to realise that "New Labour", while left of the Democrats, would still have most of the Democrat's rank-and-file.
Lib Dems would have moderates or maverickish lefties. Conservatives would be the non-nutcase proportion of the Republican party. The nutters would be in the USIP.
What each main leader would be in:
Obama: Labour
Biden: Labour
Clinton: Labour
Reid: interesting...could be any of the Big 3.
Pelosi: Labour
Romney: Conservative
Bohnner: Conservative
McConnell: Conservative or perhaps USIP.
Add some more.
It's difficult to explain just how right-wing a country America is. Most Conservative Democrats (the right-of-Bill Clinton type) as well as the non-tea party Republicans would comprise the Conservative Party. The Tea Party would be USIP. The "New Democrats" who aren't seen as "Conservative Democrats" or as "Progressives" (in the American sense) as well as plenty who actually are seen as "Progressive" would be the Lib Dems. Labour would be the "Progressive Democrats" who are seen as super left-wing here, and it would probably also include the Democratic Socialists of America, Bernie Sanders, etc.