"I cannot stress enough how out of character this is for him," said Mulcahy. "We have been together for five years, he wouldn't do this to me, we have a very happy relationship we were both looking forward to going to Bath today. I'm terrified something has happened."
Somerset Live said the couple had moved from New Zealand to Bristol in January and that friends had now travelled from London to help search for him.
Day is described as 5'11'' tall, with tattoos on both arms and was most likely wearing a cap and long wax black jacket.
Day's friends and family, including Mulcahy, have also taken to social media to express their concern and say his disappearance is out of character.
"This is extremely out of character for him, we always keep in touch every minute of the day," Mulcahy wrote on Facebook.
"Everyone who knows Joe loves him very much, he is incredibly special to me, and his family and friends."
In an email to Bristol24/7, Joseph's sister Olivia said: "Please help us find my brother – Joseph Day – a missing New Zealander in Bristol. Joseph has been missing for 35+ hours and is deemed low priority by the local police but he is NOT.
"We feel helpless overseas and can't stress how out of character and HIGH RISK this is. What if you were told your loved one wasn't a high priority and is 'low risk'?? PLEASE help us put pressure on the police to prioritise this."
A number of his former TVNZ colleagues including Breakfast host Jenny-May Clarkson and reporters Kimberlee Downs and Andrew Macfarlane have posted on social media, also urging help.
🚨If anyone's in the UK/Bristol and has seen Joseph - please get in touch with his partner Kelsey (+447851819731).🚨