Oh wow. In 1985 Doctor Who was put on hold for 18 months, sending a worldwide pre-Internet outrage and they did what was “the thing” of the time: “Let’s make a ‘We Are The World'” song! That’ll fix everything!
They *did* end up bringing it back on the air for a couple of years so perhaps it worked, only to be cancelled entirely.
The cheesy factor is not only valid for today looking back to 1985 but they knew it was cheesy even then. Colin Baker says he regretted performing in it, although I think he secretly is proud of it.