Though I don't have a graduate degree, I have original (and rigorous!) mathematics and have been cold-emailing some professors. Unfortunately, none of them want to collaborate online with me.
I really would like to get my research accepted. What can I do?
Read enough math papers so that you have a sense about how they are written up. Then write your own, emulating the style. Then submit it to a journal. You don't need any degree for this.
You will get feedback. If the work is not very good, it will be immediate feedback and you need to think about what you are doing. If it is good, then, after a while you will get a reviewer report suggesting changes.
You don't need collaborators, but finding someone to read your work and give you some feedback is helpful. If you have an undergrad degree, or are an undergrad, one of your professors can probably help.
Possibly you don't want to really ask for "collaboration" as much as "advice" or "feedback". Advice and feedback are "limited" responsibilities, while collaboration is an on-going thing. People might be willing to give advice but not make any long-term commitment.
Also, keep in mind that most professional mathematicians, by their own design, are already "fully-booked" for the foreseeable future, apart from possible large surprises which would justify changing their schedule abruptly.
That is, they already have many on-going projects, as many collaborators as they'd want (if not too many!), as well as other obligations. So, imagine that you are asking a person who is already extremely busy to do something extra. Not that that's impossible, but "it's a big ask". :)
EDIT: ... and, anyway, as in other comments and answers, you don't have to have collaboration of a professor to attempt to publish (in the old sense of "in a refereed journal") your work. Yes, the stylistic expectations, and judgement about level of innovation and so on, will be easier for an experienced person, but in that regard all you really need or want is "advice" rather than "collaboration".
Against some of the other answers, I will give a more pessimistic view here.
It will be almost impossible for you to get published from outside academia. I'll outline a few reasons.
Nothing prevents you from submitting your work to a journal; an academic affilitation is not a requirement. That said, to get published you need to impress one or more experts (the editor(s) and the referee(s)). Unless there is some crazy exceptional idea in your work, badly written mathematics will fail to impress. Which brings me to the next point.
It usually takes more time than a full Ph.D. for a mathematician to start writing mathematics decently. That is, years of practice and mentorship; a big part of supervising graduate students has to do with getting them to write properly. So I would deem it impossible that whatever results you have are written in a way that a referee will find acceptable. Refereeing a paper is a lot of work, and referees get understandably upset when papers are written poorly.
Without mentorship and contacts in academia, it would be really hard for anyone (not just you) to assess the quality of your own research. Most non-trivial research benefits greatly from discussions with other experts. And it is a common experience for all (most?) mathematicians to have a "brilliant" idea, that immediatly becomes stupid when we explain it to another mathematician. We have all thrown some "earth shattering" paper to the trash when we finally realized it was worthless.
Which brings us to the issue of contacts. Even for an absolutely brilliant mathematician, it would be hard to be heard from outside the community. Most famous case being that of Ramanujan. His first attempts to get in touch with British mathematicians ended up in rejection precisely because his (brilliant) arguments were poorly written. And some were wrong, as most of his results on prime numbers were. Hardy's wisdom and mentorship were required to make Ramanujan a mathematician. Another famous example is Fourier, whose papers were (rightfully) rejected even though they had brilliant ideas.
The only chance you have to get an inside is to hopefully get in touch with someone who is a direct expert in the (sub) area your results are in. That said, we are all used to receiving papers from cranks; the only way I would (maybe!) take a closer look to a paper sent to me out of the blue, would be if it makes interesting claims in topics I have worked myself. Even then, most people are usually busy enough with their own projects, students, classes, and service work, that they won't have the time to pay attention to you.
So, this depends to some extent on who you are asking and what you are asking, and what your results are.
First, let's be clear that people aren't likely to to be interested if you have a claimed proof of some famous open problem, like the Riemann Hypothesis, or P != NP, or the Collatz conjecture.
Second, asking people out of the blue to collaborate is in general a not at all small ask. If you are doing so, you need to at a minimum explain in your email a) what result you think you have b) how it works and c) what part requires collaboration. Is there a specific technique or aspect that they are an expert on? For example, if someone sent me an email asking to collaborate and it was on something I don't know much about, I wouldn't respond positively.
If you have a result that doesn't need someone else, then write it up, and then talk to someone who is an expert on a closely related problem. After you have their feedback, then send it to a journal if it is good and worth looking at.
What you are asking for here is for someone to do the job of a PhD advisor but to do it for free and without getting any credit for it at their work. If you want a PhD advisor you're going to have to apply to a PhD program.
The only likely exceptional circumstances here would be:
Another idea is to pay them. A university lecturer or professor normally gets paid to help students get published - and it's a full time job. Why would they do the same work for you for free? But lecturers also often offer for-a-fee tutorial services to students, mainly for those who need extra help to catch up. But a maths lecturer offering such services may be willing to tutor somebody through the process of writing up and submitting a paper instead. It might make an interesting change of pace for them. And if in the process they end up reading your work and having ideas of their own about it, that could conceivably turn into collaboration.
But don't get your hopes up too high. It's hard for outsiders to judge what's of sufficient interest to be worth publishing, and it's particularly hard for outsiders to overcome the credibility barriers raised as a result of the thousands of amateurs who seek instant fame for solving famous problems, who turn out to be fooling themselves. You should take seriously the possibility that you might end up paying someone a lot of money, only for them to tell you that the work is unpublishable, or to see your months of hard work being rejected by the journals. Such is the academic life. If that's going to upset you, don't start down this road.
As others have already said, the best way to get into mathematical academia (or any mathematical career) is by going through a postgraduate programme - this is exactly what they are for! And you are competing with everyone already on that path. Attempts to find a faster shortcut are always going to be more difficult and less reliable than following the main highway.
There are some good answers already. Let me add
As to what to do,
