My first thoughts on reading your initial posts was that there was a lot of backstory to this.
The scenarios I thought of were - is this a new team that you are responsible for setting up? (start as you mean to go on, but do you need to establish relationships within the team?) Is this a team functioning well that you are now responsible for? (don't change things that are working already) Is this a team that doesn't work well and you are being asked to fix things?
Since you have clarified that it is more of the latter situation, that influences things. A team that is in some ways dysfunctional will not necessarily cope with "for goodness' sake talk to each other like grownups and sort things out", even though that seem reasonable. If it was a smaller group then frank talking like that on some issues might be appropriate to try, but 14 is too many. And now their over-the-top straight-to-the-union response tells you more about what you are dealing with.
So, where to go from here? I think you need to find out from all of them what is working, what is not, what the problems are - from their perspective. At the same time you can be defining what problems you see yourself, because yes both sets of problems will need sorting out.
My suggestion, if you haven't already done something similar, is that you need to prioritise individual meetings to find out what each member of your team wants and needs, what they think are the urgent problems, what they like and want to stay the same. I'd try to do those meetings within as short a period as possible, and focus on listening yourself (don't give anyone any ammunition to complain again). Then sit down and create a brief summary of what you think is going on, with possible solutions based as far as possible on ideas others have mentioned.
After all that, have another team meeting. Feed back "you have all told me that ..." and "some great suggestions on dealing with it are ..." And then see what happens?
I don't know enough about how discussions with union reps are supposed to work. If you don't either then seek urgent advice on that specifically. It might be, for example, that a discussion with the rep after your individual meetings and before the team meeting, would be appropriate and helpful.
Good luck!