An apparatus comprises at least one processing device configured to identify, utilizing virtualization software running on a storage system, a snapshot lineage comprising snapshots of a storage volume, the snapshot lineage comprising (i) a local snapshot lineage on the storage system and (ii) a cloud snapshot lineage on cloud storage external to the storage system. The processing device is also configured to utilize the virtualization software to select a given snapshot in the cloud snapshot lineage to recover to the local snapshot lineage, to expose a cloud storage volume for the given snapshot, and to map the cloud storage volume to a target local storage volume on the storage system. The processing device is further configured to recover the given snapshot to the local snapshot lineage by utilizing a data mover of the storage system to write data of the given snapshot to the target local storage volume.