The storage locker service touched down in the US last year, and has now debuted in Britain with 5GB of free storage on offer.
Premium subscribers can gain access to 20GB of data for £6 per year, while those who require a significant of space can purchase 1TB for an annual £320.
Companion apps for Mac and PC will help users organise their files, while the Cloud Player for mobile devices is used to run media files online.
The launch of Cloud Player in the UK puts Amazon in competition with rival services iCloud, Google Drive, Dropbox and SkyDrive, which all offer free storage as well as premium packages.
Amazon is expected to announce new hardware at its event on September 9. The Kindle Fire 2 has been tipped for unveiling that day, and the launch of Cloud Drive in the UK could suggest the tablet range is about to make its long-awaited debut in the territory.