Not every mom is willing to go to a psychiatrist, sadly. Especially if she already thinks it's all "bullshit". Also, depending on where you live, how well educated said psychiatrist is on nonbinary issues can vary. If you plan to go about it that way, make absolutely sure that said psychiatrist believes nonbinary is even a thing and that they will stand on your side. Otherwise you'll end up with two people trying to tell you that your feelings are wrong and attempt to push you back into cisnorms.
I'm not trying to scare you away from the suggestion. I'm just telling you this because of experiences I and many of my trans or nonbinary friends have had. There are good psychiatrists out there that will make your life so much better, and then there are really shitty ones. I don't wanna go too much into detail, but mine almost made me kill myself and I am not the only one. As I said, I don't wanna scare you away from this option if you do find a good one, I'm just saying you should be careful because the last thing you'll need in the coming out process is for your psychiatrist to turn against you.
If you can find a good psychiatrist that you can trust, and if your mum is willing, then I would definitely suggest doing that.
If you can't find a good one, or if your mom is extremely reluctant or doesn't respect a psychiatrists opinion, then don't do that. It can actually make the situation worse.
Regardless, I would suggest continuing to express yourself the way you prefer and standing firm, but not being too pushy. By this I mean, correct people on pronouns, wear whatever you like, but don't start arguments or lecture them on it. Just be you and let them work it out themselves. A lot of people think something is just a dumb phase when they first hear about it, but if you remain consistent in your identification it will become undeniable to them. Society is changing too, so if your mum can be changed then she will with time if you stand your ground.
Some parents also need a mourning phase. Sounds dumb, but my mom is very lgbt-friendly and even she needed a couple of months to mourn the "daughter she lost". My mom was honest about needing this, but many other parents go into denial to avoid having to enter this process. Heard about the stages of sorrow? That one is relevant here. If she is the type to go into denial, you may also have to expect a stage of rage, among others. As I said, stand your ground.
There is an exception though. If your mum is fundamentally homophobic or transphobic and you still depend on her financially, I would suggest to wait with ALL of the above until you are able to move out and support yourself. There are too many risks involved with angering someone you depend on if they straight up hate the LGBT community.
You can also wait with coming out or do it more slowly if you are uncomfortable with going all in. I know I attempted to come out online and was met with so much anger from my then close "friends" that I went quiet for a while. Then I came out online to people who didn't suck, then in school, then to my family. The last two happened pretty quick though.