The former employer has a legal obligation to provide your daughter with a P45 but I'm unaware of a specific time-limit. (P45s are usually issued with, or just after, an employee's final pay is given to them).
However, if your daughter only has a part-time job it's unlikely that her earnings will reach the threshold for paying tax anyway, so (in the absence of a P45) simply completing a P46 for the new employer should suffice anyway.
It is a bit odd in that it is framed as a Freedom of Information Inquiry and answer which I wouldnt have done: I just would have written.
EVEN if the employee has not struck the tax threshold I would have thought a P45 is needed or else *gasp* she would be underpaying income tax in the next employment.... dear dear