The song, by the band Fat Les, became a football anthem during the 1998 World Cup.
They are urging the public to help get the track to number one and to dig deep to buy Vindaloo Two t-shirts.
Mellor had been teasing the track on social media in recent days, sharing little clips of the stars taping the song.
He has now posted a video on Instagram encouraging fans to help it hit the top spot.
“Come on, let’s get this song to number one,” he urged. “Not only to say thank you to the NHS, but also to get behind England in the Euros!"
He went on: “Come on people, put this on before every England game, put the song on, listen to it, get into the spirit and gather your family around.
"We can do this!"
“Come on England, it’s our time," added the star.
Watch: Vindaloo Two video
The video - which sees the group joined by joined by the London Community Gospel Choir and NHS workers - has gone down a storm with fans.
"It is awesome," gushed one person on Instagram, while another said it put a smile on their face.
"Just watched this and it’s brilliant, pure joy right there," said another.
Watch: Will Mellor talks about how much his new role means to his mum