Hello!
I would say that the first option is less clumsy and flows better because the main point that you are trying to make is that your colleague will get in touch in due course. The comma after course allows for a pause and indicates to the reader that the next part of the sentence is "extra information" - in this case why his colleague will be getting in touch in due course.
1) My other colleague will get in touch with you in due course, to finalize the report.
Just something else that I noticed: Unless you have been referring to another (or second/third) colleague in this sentence you can leave out "other".
1) My colleague will get in touch with you in due course, to finalize the report.