  Context. Galactic cosmic rays are widely assumed to arise from diffusive
shock acceleration, specifically at shocks in supernova remnants (SNRs). These
shocks expand in a complex environment, particularly in the core-collapse
scenario as these SNRs evolve inside the wind-blown bubbles created by their
progenitor stars. The cosmic rays (CRs) at core-collapse SNRs may carry
spectral signatures of that complexity. Aims. We study particle acceleration in
the core-collapse SNR of a progenitor with initial mass 60 $M_\odot$ and
realistic stellar evolution. The SNR shock interacts with discontinuities
inside the wind-blown bubble and generates several transmitted and reflected
shocks. We analyse their impact on particle spectra and the resulting emission
from the remnant. Methods. The hydrodynamic equations for the evolution of SNR
inside the pre-supernova circumstellar medium have been solved simultaneously
with the transport equation for cosmic rays in test-particle approximation and
with the induction equation for the magnetohydrodynamics (MHD) in 1-D spherical
symmetry. Results. The evolution of core-collapse SNRs inside complex
wind-blown bubbles modifies the spectra of both the particles and their
emission. We have found softer particle spectra with spectral indices close to
2.5 during shock propagation inside the shocked wind, and this softness
persists at later evolutionary stages. Further, our calculated total production
spectrum released into the interstellar medium demonstrates spectral
consistency at high energy with the galactic CRs injection spectrum, required
in propagation models. The magnetic field structure effectively influences the
emission morphology of SNR as it governs the transportation of particles and
the synchrotron emissivity.
