It is best to check the processing times for form I-539 for the exact time frame. But typically it takes about 90 days to get the H4 processed. If filed along with the H1B case, H4 cases are processed at the same time. So if you file the H1B premium processing along with the H4 request the case can be processed in 15 days.
But in your case, the H1B is already approved so you are looking at 90 days more or less.
Estimated processing times are as posted on the USCIS website.
J Charles Ferrari
Eng & Nishimura
The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship.
Attorney Ferrari has provided an accurate comment. You need to use the USCIS site for the local times for processing as it varies in different regions of the US.
Legal disclaimer: The statement above is provided by CC Abbott is based on general assistance and not intended to be a legal opinion because not all the facts are provided. The person requesting information and all others reading the answer should retain an attorney who is permitted by the state bar within the jurisdiction who can examine the complete facts and provide a legal opinion on your case. All information provided in the above answer and other information provided by CC Abbott does not create an attorney/client relationship within any state of Federal law.