Watts was born in Shoreham, Kent, England, the daughter of Myfannwy "Miv" (née Roberts), an antiques dealer and costume and set designer, and Peter Watts, a road manager and sound engineer who worked with Pink Floyd (her father's manic laugh and mother's comment about "cruisin' for a bruisin'" are featured in the song "Money" from The Dark Side of the Moon). Watts is pictured, in her mother's arms, with her father, brother, the band, and other crew members, in the hardback edition of drummer Nick Mason's autobiography of the band Inside Out.

Watts has one brother, Ben, a year older and now a photographer in the United States.

Watts' parents separated when she was four years old, and when she was seven, her father died. Following her father's death, her mother relocated the family to Llanfawr Farm, on the Isle of Anglesey in North Wales, where they lived with Watts' maternal grandparents, Nikki and Hugh Roberts.

Watts described her mother as a hippie "with passive-