The usual approach involves interchanging the position of the column you want removed and the last one, and returning the Cholesky to triangular form via either Givens rotations or Householder transforms, whereupon you can simply drop the last one.
(Similarly you can update an additional column into any position by adding to the end and then swapping into the desired position followed by the Givens/Householder transformations to fix it back up to triangular form.)
When you do the swap of columns, you destroy the triangular form you need it to have. Each of the Givens rotations (or Householder transforms) then fixes up one of the values that are 'sticking out', until they are all fixed. This is quite a lot faster than recomputing from scratch.