Henri and Daisy met at a dinner party in London, where he was working at the French embassy and she was studying at the London School of Economics. He had planned to leave this party as soon as he could, because he had heard there would be a Danish princess there and he was sure she would be very boring. Then he got there and found himself absolutely smitten.
There wasn't really any opposition - the King said yes right away, and as far as I know the Danish people thought it was all very romantic. Henrik officially converted to Lutheranism before the wedding, so no issues there, and the nationality was fine as long as it wasn't Danish. He was very much just a commoner, by the way - there was an unsuccessful attempt to ennoble his family a couple centuries ago, but in the end it didn't happen.