Compact rolled-up Co–Cu nanomembranes of high quality with different numbers of windings are realized by strain engineering. A profound analysis of magnetoresistance (MR) is performed for tubes with a single winding and a varied number of Co–Cu bilayers in the stack. Rolled-up nanomembranes with up to 12 Co–Cu bilayers are successfully fabricated by tailoring the strain state of the Cr bottom layer. By carrying out an angular dependent study, we ruled out the contribution from anisotropic MR and confirm that rolled-up Co–Cu multilayers exhibit giant magnetoresistance (GMR). No significant difference of MR is found for a single wound tube compared with planar devices. In contrast, MR in tubes with multiple windings is increased at low deposition rates of the Cr bottom layer, whereas the effect is not observable at higher rates, suggesting that interface roughness plays an important role in determining the GMR effect of the rolled-up nanomembranes. Furthermore, besides a linear increase of the MR with the number of windings, the self-rolling of nanomembranes substantially reduces the device footprint area.