I converted mine a couple of years ago. I simply walked into FCL at the Belgrano with my licence and logbooks in hand. They gave me a very simple form to fill in, had a look at my logbooks, took a photocopy of my latest LPC and JAA medical and gave me everything else back.
I then paid £168 and walked out 25 minutes later. The service was swift and courteous. The new JAR-FCL licence appeared in the post within a week.
It might take a bit longer if you only barely meet the requirements.