Three is the first network to offer the operating system upgrade to owners of the handset, announcing via Twitter that it has begun distributing it over the air.
Users can manually check if the update is available by accessing the 'About device' section in the 'Settings' menu, but it is likely to be several days before all Three users receive it.
O2, Vodafone, Orange and T-Mobile are expected to begin distributing Android 4.1 to Galaxy S3 users soon, though official announcements are yet to be made.
Samsung initiated a gradual roll out of the operating system in Europe last month, with users in Poland being the first to report it landing on their devices.
Jelly Bean was released along with the Google Nexus 7 tablet over the summer. The new operating system offers smoother navigation, improved voice support and the Google Now personal assistant.