Fathom is an agent with CELTS.
At least she is in the first book. In the second, unpublished, and likely unfinished, it is probable that the author intended she would continue to work for that organization but we will never know for sure and the adventure depicted in the movie made based on the second escapade makes no mention of CELTS or Fathom being an agent for anyone.
But in the first book she is decidedly an agent, well trained and given a specific mission to perform. That does not mean she is a voluntary operative, though. Her recruitment came when she was nabbed shortly after killing the SOB who had gotten her hooked on heroin and loaned her out to numerous friends. "Work for us or go to prison" was the word from the CELTS leader at the time.
The question was presented after she had gone through a very rough cold-turkey weaning from the addiction she had, complements of a nasty Hollywood producer who had wooed her to L.A. and then systematically gotten her hooked on smack. After many months of abuse and being an unwilling plaything, she put several bullets into the bastard and fled. CELTS agents were waiting for her.
Before her ill-advised trip to the La-La-Land, Fathom, born Jane Elisabeth Harvill, born in Sweden though a British subject, was a 22-year old daughter of a British diplomat. She was raised all over Europe as his father's foreign service took him to posts throughout that continent and she grew up speaking several languages as a native and a couple of more passably. She was an intelligent, energetic, happy young woman whose one mistake was believing a glib talking charmer.
After she dispatched her abuser, Fathom's life took another drastic turn and she began a career as an operative for "Counter-Espionage (Long Term Security)", a British-American organization tasked with looking ahead to problems that were on the horizon and stopping them before they grew.
She would become a very good agent for CELTS, assuming she lived long enough.