How long it takes varies by school. It can take anywhere from a couple days to a couple months, depending on the processing time for the particular school. In addition, you will need to follow the rules the school has for PLUS, so it might involve more than just applying - and the application itself has to be "just right" - no mistakes (you would not believe how many social security number, birth date, missing amounts, missing terms, students who obviously signed for their parents, etc financial aid offices see).
How do you know that your mom was approved? Did she do the online application? If so, do you know that your school uses that? The school where I work does not participate, so parents must apply with a paper application & that has to be processed.
If your school participates in the online application and your mom has been approved, has she signed the Master Promissory Note yet? She would need to go to www.studentloans.gov
to do that - she will have to sign in with her FAFSA PIN. Is the PLUS loan in her name or in your dad's name? The PIN used to sign the MPN must belong to the person on the PLUS application.