OP, I think you know you are right when you say that you can only control what you do, not anyone else. No one else is going to step up, and no one else will be able to set your boundaries.
I had a relative like this, many years ago, who would "emotionally dump " via email. We were quite close, so I felt obliged to read everything just in case there was actually something of substance in the emails, or something urgent. I'm talking about pages and pages of rambling worries, six or seven times a day, from another country, almost an entire working day ahead of me, due to the time difference. There was absolutely nothing I could have done, even if there had been a problem.
This went on sometimes for weeks, until my relative apparently either solved his issue or stopped worrying about it. Then it would start all over again.
For a few years, this happened all the time, and stressed me out to the point that I knew I had to set some boundaries. I finally asked a really good friend to "triage" the emails. I forwarded every single email directly to her, she would read them, then let me know if there was anything at all urgent.
Do you know what? There never was. All he was doing was taking all his stress and worry, typing it out into an email, then hitting send. Away goes all his stress, dumped onto me! It was his way of coping. Not good for me!
So, I stopped reading his emails, in fact set it up so they went directly to a spam folder. The world did not end. Once a month, we would talk by phone, and never once did he mention any huge problems he had.
Your uncle is emotionally dumping on you. His problems are not yours to solve.
I would message his sons and your mother and say, in no uncertain terms, that I would no longer be answering the phone or reading messages from your uncle (you don't have to give a time frame), and if anything happens to him, it's on them.
Then mute, block, and if you want to, call your uncle once a month to chat and see how he is. Don't unblock him, don't start looking at messages again, just chat if you want to. My guess is that he will have moved on to someone else.
I know it seems harsh, but by setting boundaries, you can talk to him on your own terms. You cannot be responsible for what your mother or his sons do, so don't even bother getting involved with that.
Not your circus, not your monkeys.
ps. forgot to say that people with cancer are just people who may be having a difficult time with their health ( I know, I was one of those people), but it doesn't mean they are better than anyone else. You have a right to a healthy, stressfree life, as well.