If you have an Amazon Fire TV stick, it may possible to cast to it from either an Android or iOS mobile device, but you will need to install a 3rd party app on both the Amazon Fire TV stick and your mobile device. (Examples include Video and TV Cast for Android and iWeb TV for iOS). These apps act like web browsers, enabling you to search for voting.bafta.org and then login to it before selecting your film. Once selected, the video will be scanned by the app and is ready to beam to the receiving version of the app on your Amazon Fire TV.
It is important to emphasise this cannot be guaranteed to be 100% successful as it is dependent on the compatibility and any restrictions on the video files. BAFTA cannot be guarantee the effectiveness of, or give support for, third party apps. Wherever possible, we therefore recommend using one of the other methods available – such as HDMI cable connection.