Kensington Palace opened earlier this week a Twitter account for the Duke and Duchess of Cambridge and Prince Harry of Wales, meant to cover their activities as well as updates on their Royal Foundation.
The link to their Twitter account is https://twitter.com/KensingtonRoyal. Buckingham Palace's Twitter account can be found at https://twitter.com/BritishMonarchy, while the activities of the Prince of Wales and Duchess of Cornwall are covered by https://twitter.com/ClarenceHouse.
The Duke of York is also on Twitter: https://twitter.com/TheDukeOfYork. Tweets sent by HRH himself is signed - AY. I doubt that the Duke and Duchess of Cambridge will tweet themselves, though.
Updated on Friday 1 April 2016 at 09.00 (typo corrected).