Calcium phosphate (CaP)-based ceramic–biopolymer composites can be regarded as innovative bioresorbable coatings for load-bearing implants that can promote the osseointegration process. The carbonated hydroxyapatite (cHAp) phase is the most suitable CaP form, since it has the highest similarity to the mineral phase in human bones. In this paper, we investigated the effect of wet chemical preparation parameters on the formation of different CaP phases and compared their morphological and structural characteristics. The results revealed that the shape and crystallinity of CaP particles were strongly dependent on the post-treatment methods, such as heat or alkaline treatment of as-precipitated powders. In the next step, the optimised cHAp particles have been embedded into two types of biopolymers, such as polyvinyl pyrrolidone (PVP) and cellulose acetate (CA). The pure polymer fibres and the cHAp–biopolymer composites were produced using a novel electrospinning technique. The SEM images showed the differences between the morphology and network of CA and PVP fibres as well as proved the successful attachment of cHAp particles. In both cases, the fibres were partially covered with cHAp clusters. The SEM measurements on samples after one week of immersion in PBS solution evidenced the biodegradability of the cHAp–biopolymer composites.