Aside from Harry Potter, comedian and actor Tavaré also co-wrote and starred in Bafta-winning ITV series The Sketch Show.
He is next due on screen in upcoming movies The Magnificent Room and Cockroaches, and was scheduled to take part in a stand-up tour in the spring.
Friends and fans have sent their support, with hundreds of messages wishing him well.
One wrote, “My thoughts are with you both and everyone affected. Stay strong, big man,” while another added: “Thanks for letting us know Laura bless you both with the strength to get through this. Sending you both positive thoughts.”