Does this lead to ambiguity? If you say I will travel the first week in May can that be misconstrued to mean the first Monday in May?

Where there is any fear of misinterpretation, hari (day) is added. So: minggu depan (next week) hari Minggu depan (or more colloquially besok hari Minggu) next Sunday.

Generally speaking it seems to cause less confusion than the perennial problems associated with the English word 'next' in time expressions.