A dancer named Melissa Hayden performed the dance scenes, as the
following source indicates:
Turner Classic Movies
"But 'Limelight' was primarily a one-man show with Chaplin writing,
producing and directing the film. He composed the lyrical score,
co-wrote the comedy songs and even choreographed the dance numbers.
The 'Death of Columbine' ballet number was originally conceived by
Chaplin in 1950 for Alicia Markova and Anton Dolin. When filmed for
'Limelight' the roles were performed by Andre Eglevsky and Melissa
Hayden (who doubled for Bloom in all Thereza's dance scenes)."
This is confirmed here as well:
"... my Aunt Melissa (Melissa Hayden) had been in the film as Claire
Bloom's double and had helped to choreograph the movie along with her
NYC ballet partner Andre Eglevsky."
I hope you aren't disappointed to find out the answer!
I used the following search term:
"claire bloom" limelight ballet double
Then I saw the reference to "Claire Bloom's double" in the Cally
story, then search on Melissa Hayden's name.