If you want to join BA they won`t take you unless you have a UK ATPL - trained in the UK too I think. (well obviously, or it wouldn`t be a UK ATPL)
Wrong, many BA pilots have trained outside of the UK. Oh, and "well obviously, or it wouldn't be a UK ATPL" is also wrong. One could have done the CPL in the UK, the IR in, say, Portugal, and it would still be a UK issued ATPL. The licence does not say in which state a rating has been added from, as it is ultimately the UK CAA that endorse the licence with ratings.
To the OP, transferring EASA states basically involves the transfer of medical records to the new state, and the issue of a new licence from the host state, finally, the cancellation of the first EASA licence. The process can take up to three months, depending on the states involved.