Formal & Official:
* Properly
* Appropriately
* Legally
* Officially
* Formally
* Correctly
* Accurately
* Regularly
* Punctually
* Timely
* In due course
* As required
* In accordance with
* According to
Informal:
* Rightfully
* Deservedly
* Properly
* Completely
* Thoroughly
* Entirely
* In good time
* With due diligence
The best synonym will depend on the specific sentence and the meaning you want to convey.