1-They will take the matter up again later on next month.
2-They will take the matter up again later on, next month.
Does the comma change the meaning in any way?
It does not seem to me that it does. In either case they will take the matter up next month. I am not sure that there is a point in having 'later on'. One could think that in '2' 'next month' has been added as an afterthought.
In the first case, the "taking up" will happen in the latter part of the next month. "Later on next month" is one idea.
In the second, the "taking up" will happen later on. This "later on" will take place at some time in the next month. Could be the first day of the next month.