  We consider the twistor description of classical self-dual Einstein gravity
in the presence of a defect operator wrapping a certain $\mathbb{CP}^1$. The
backreaction of this defect deforms the flat twistor space to that of
Eguchi-Hanson space. We show that the celestial chiral algebra of self-dual
gravity on the Eguchi-Hanson background is likewise deformed to become the loop
algebra of a certain scaling limit of the family of $W(\mu)$-algebras, where
the scaling limit is controlled by the radius of the Eguchi-Hanson core. We
construct this algebra by computing the Poisson algebra of holomorphic
functions on the deformed twistor space, and check this result with a
space-time calculation of the leading contribution to the gravitational
splitting function. The loop algebra of a general $W(\mu)$-algebra (away from
the scaling limit) similarly arises as the celestial chiral algebra of
Moyal-deformed self-dual gravity on Eguchi-Hanson space. We also obtain
corresponding results for self-dual Yang-Mills.
