The process you follow is the usual endorsement process, so it's entirely dependent on how long the endorsement queue is currently. Last I heard, it's pretty quick these days - more like a few days rather than a few weeks. However, I would look to manage your employer's expectations appropriately and under promise / overdeliver.
When you go through the CCSP endorsement process as an existing CISSP, the endorsement form identifies you already hold the CISSP so will mandate that ISC2 acts as your endorser. Accordingly, it will remove the section of the form where you usually enter details about the endorser. Also, as the CISSP acts as a full experience waiver for the CCSP, the form will also remove the section where you usually enter experience information. Aside from those things, it's really no different to a typical endorsement application - certainly in terms of timeframe.
Coming back to the point about the CISSP acting as a full experience waiver for the CCSP, it really is time to stop that. There are now more than 10,000 members who hold the CCSP, and the certification is six years old. So, it really should be able to stand on its own two feet by now.
Also, as it now qualifies for IASAE III level roles, I can't see how it's allowed for someone who potentially has zero cloud experience to be granted this certification just because they are a CISSP and have passed the CCSP exam.
Anyway, good luck with your exam!