If you wish to operate using JAA class 1 medical certificate privileges on aircraft registered outside the UK you would need to obtain special permission from the appropriate regulatory authority.
Ok that's basically saying that you can't use your JAA Class 1 Medical to fly foreign aircraft unless approved by the registry authority.
In most cases when flying foreign registed aircraft you will require the medical appropiate to that country.
Some countries verify foreign medicals upon completing a foregin verificaiton process. (Often done in conjunction with a PPL verification to exercise your PPL based on the JAA PPL.)
Therefore, if you were to travel abraod to complete CPL training on your JAA Class 1 Medical, it
might (Depending on whether you try to apply for acceptance) not be accepatable. However, you can apply for a medical in that country.
In the US for example a FAA Class 3 Medical will suffice for training and would only cost under $100 (estimate). You could complete your JAA CPL and test and then have your medical deviation removed.
PM if you are confused still.
Hope this helps, and of course good luck!