I was wondering if there is a characterisation for $C$*-algebras (unital) for which the bidual does not have any central atoms.

It is not sufficient for example to demand that the $C$*-algebra does not have a non-zero commutative algebra direct summand:

Consider the $C$*-algebra $$A:=\bigl\{f\in C([0,2];\mathrm{M}_2(\mathbb{C}))\colon \ f \mbox{ is diagonal on } [0,1]\bigr\}.$$

Then $A$ does not have a non-zero commutative algebra direct summand, but the atomic part of the bidual should be $$A^{**}_{atomic}=\bigl\{f\in\ell^\infty([0,2];\mathrm{M}_2(\mathbb{C}))\colon f \mbox{ is diagonal on } [0,1]\bigr\}$$

which has plenty of central atoms. See Commutative direct summands of C*-algebras.

It is true that the bidual of a $C$*-algebra $A$ has no central atoms if and only if the state space $S(A)$ of $A$ has no singleton split faces. This is Corollary 5.33 and Corollary 5.34 in Geometry of State Spaces of Operator Algebras by Alfsen and Shultz. This characterises these algebras in terms of the state space, but knowing a condition on $A$ would be very interesting.