Stevegg5:
You are absolutely correct. I have been through the same process a few months ago. Just note that for the issuance of a first JAA/EASA license, it used to be mandatory for the skill test to be observed by a UK CAA inspector. This requirement is no longer, however they still can decide to do so. Your TRE must inform the UK CAA that he will be conducting a skill test for the issuance of an ATPL/type-rating issuance and that it will be for a conversion from an FAA ATPL. Normally, the UK CAA will decide not to attend the skill test. Once the skill test is done you need to submit all the paperwork to the UK CAA office in Gatwick with all you original logbooks. They will keep them and you can expect them to be returned together with your new license within a few weeks. Also, note that you will need a current EASA medical for the license to be issued.
Hope that helps and all the best