Geeson was born in Arundel, Sussex and moved to London at the age of ten. She came from a middle class family, and her father edited the National Coal Board magazine. Geeson attended Corona Stage Academy, making her stage debut in 1957.
In 1984, Geeson left London for Los Angeles where she decided to stay. She was married to the actor Kristoffer Tabori from 1985 until 1989. Among other roles she appeared regularly on hit US sitcom Mad About You as neighbour from Hell, Maggie Conway. Having always had an interest in antiques, she now runs her own antique store, Blanche & Co in Beverly Hills in addition to her acting work.