Not sure where to find a source on this, but here is my opinion as a native speaker:
propose are definitely synonyms.
First, I find one subtle difference between them is the amount of confidence the speaker/writer has in what is being offered.
Just between the two,
put forward shows more confidence than
(strongest) I will push forward with the proposal at the next meeting.
(strong) I will put forward the proposal at the next meeting.
(weaker) I will bring forward the proposal at the next meeting.
(weakest) I will [suggest / offer] the proposal at the next meeting.
So when I read your example sentences, the one that uses "bring forward" makes me think that the proposal is more open for discussion and not yet complete. But the one that uses "put forward" makes me think that it has already been considered thoroughly and less discussion or opposition is expected.
Another subtle difference is whether the proposal still needs to be made or not. When I read your example sentence:
United Nations must [bring forward / put forward] a proposal to denuclearize South Korea to secure World peace.
"bring forward" carries a hint that the proposal still needs to be made.
"put forward" carries a hint that the proposal is already made.