I want to clear this up for everyone ahs no one has mentioned it as clearly as the following statement.
"The CPL must be completed with a FTO (Flight training organisation) under the same civil aviation authority as the ATPL theory exams were completed under."
I.e if you take your exams under the UK CAA you must do your training at a UK CAA FTO - that does mean you can train in the USA, Australia etc just must be under a UK approved school.
The IAA and CAA have recopricol agreements. CAAs may also provide exemptions but this most be in writing to be effective.
Section A8 of LASORS at
LASORS 2008 | Personnel Licensing | Safety Regulation Where permission is required - you should assume this to be written permisison.