You sure can do the full JAR rating in the US! Only a British one you cannot.
Not true. The Skill Test for a JAA IR must be completed in a JAA member state and, consequently, training providers will insist that a proportion of the course (usually the last 5-10 hours) is also completed in that member state. If the initial IR Skill Test is done in the US then the rating has not been completed in accordance with JAR-FCL 1 and it is unlikely that any JAA member state will accept it; the UK CAA certainly will not.
There is no such thing as a 'British' IR, nor has there been for a considerable number of years.