ABSTRACT Supernova remnants (SNRs) are known to accelerate cosmic rays from the detection of non-thermal emission in radio waves, X-rays, and gamma-rays. However, the ability to accelerate cosmic rays up to PeV energies has yet to be demonstrated. The presence of cut-offs in the gamma-ray spectra of several young SNRs led to the idea that PeV energies might only be achieved during the first years of a remnant’s evolution. We use our time-dependent acceleration-code RATPaC to study the acceleration of cosmic rays in supernovae expanding into dense environments around massive stars. We performed spherically symmetric one-dimensional (1D) simulations in which we simultaneously solve the transport equations for cosmic rays, magnetic turbulence, and the hydrodynamical flow of the thermal plasma in the test-particle limit. We investigated typical circumstellar-medium (CSM) parameters expected around red supergiant (RSG) and luminous blue variable (LBV) stars for freely expanding winds and accounted for the strong γγ absorption in the first days after explosion. The maximum achievable particle energy is limited to below $600\,$TeV even for the largest considered values of the magnetic field and mass-loss rates. The maximum energy is not expected to surpass $\approx 200\,$ and $\approx 70\,$TeV for LBVs and RSGs that experience moderate mass-loss prior to the explosion. We find gamma-ray peak-luminosities consistent with current upper limits and evaluate that current-generation instruments are able to detect the gamma-rays from Type-IIP explosions at distances up to $\approx 60\,$ kpc and Type-IIn explosions up to $\approx 1.0\,$ Mpc. We also find a good agreement between the thermal X-ray and radio synchrotron emission predicted by our models with a range of observations.