「遅{おく}れちゃった、寝過{ねす}ごしたので。」
is a completely normal sentence in the real Japanese-speaking world. Japanese word order, phrase order and clause order are far more flexible than it appears to me that they are taught in Japanese-as-a-foreign-language.
Thus, it is an everyday occurence for certain conjunctions to be located at the very end of the sentences. You will often see/hear a sentence structure such as:
「(Result/Effect Clause)、+ (Reason/Cause Clause) + ので、から, etc. 。」
which is virtually the same in meaning as:
「(Reason/Cause Clause) + ので、から, etc. + (Result/Effect Clause)。」
Conjunctions in general are often placed at the very end of the sentences, which seems to baffle some learners. Those conjunctions would include:
が、けど、けれども、のに、なのに, etc.