Welcome back to the Wefugees Community and thank you for your question.
In general, asylum seekers (people during the asylum procedure) and people holding a Duldung receive the so-called "Asylbewerberleistungen" according to the "Asylbewerberleistungsgesetz". When the asylum process is completed successfully, people can apply for a residence permit at the local immigration office. As soon as they apply for the residence permit, they should also reigster with the local Jobcenter. People holding a humanitarian residence permit are in most of the cases entitled to receive money from the jobcenter instead of the "Sozialamt". An exception is here the residence permit according to section 25 (5) of the German Residence Act: In this case, people will only "move" to the jobcenter after 18 months.
It is important to take care of the application for the "Arbeitslosengeld II" at the jobcenter as soon as possible, in order to prevent a gap of the payings. Beside the actual application form, people need to submit a couple of documents (as the bank account statements of the past 3 months, ID, confirmation of the residence permit application, membership certificate of the health insurance, renting contract etc.). So the duration of the process actually also depends on the cooperation of the people and their speed in submitting the required documents. However, I am sure there are also variations regarding the duration of the process from jobcenter to jobcenter.
In case you are struggling with collecting the documents or you are afraid that there could be a gap of the payments as soon as you "change" to the jobcenter, you may want to contact a counselling office. I am sure they can support.
I hope this answers your questions and please don't hesitate to get back to us.
All the best,