On the countrary, nonprofits need unions more than for profits. They exploit their workers more. They have fewer resources and exploit their mission to get more work from their workers.
Yes, workers in non-profits are status-compensated as well as monetarily compensated. I don't think this is an argument for non-profit unionization.
If I'm donating money to fight cancer, and the majority of the money goes to administrative staff, that's inherently a flawed charity. It's exactly what led to the downfall of the Susan G Komen foundation.
There's also a death spiral problem. If donations drop and administrative costs stay the same, that charity's ratings only get worse.
There's a reason most examples of successful non-profit unions all rely on steady streams of government grant funding.