It's not a government campaign. It's run by a guy called Kash Singh. He trademarked it in 2005, when he was a policeman. Based in Yorkshire it does seem to be a genuine attempt to unify around common themes. This is what the website says:-
One Britain One Nation brings us together, not to focus on our differences but to celebrate the values we share: tolerance, kindness, pride, respect, and a tremendous desire to help others.
Today’s Britain boasts a wonderful array of cultures. It is our multicultural identity that makes Britain so unique.
Our diverse cultures are inextricably linked by the sole fact that we are British. It is this fact that has prompted OBON to reinforce and revive what collectively unites us.
OBON aims to give a new impetus for the creation of a harmonised society, to make Britain an international model of moral rectitude.
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The DfE tweeted re OBON day and the publicity followed.
However there is also an all party parliamentary group, I'm not clear why.
Link below.
https://publications.parliament.uk/pa/cm/cmallparty/190508/one-britain-one-nation.htm
The song was written by children.