A spokesman for the Prime Minister said: "The PM has always been a passionate supporter of animal welfare and believes in giving animals the best start in life."
Symonds revealed the dog's name on Twitter with photos of him exploring his new home.
"Congratulations! What a wonderful team! Dylan is such a lucky dog. Thank you for rescuing," a user replied on the post.
"Fabulous news! Thank you for supporting the fight against puppy farms and encouraging people to rescue. ??My little Albert says hi," another user replied.
Symonds also thanked a member of the Friends of Animals Wales (FOAW) Eileen for saving Dilyn.
"Thanks to the wonderful Eileen who rescued Dilyn after she got a tip-off that he was to be dumped by a puppy dealer because he was born with a crooked jaw, Eileen fixed his little jaw and saved his life. She is a hero." Symonds said.
FOAW is a small volunteer-run rescue shelter based in the valleys of South Wales.