I will be moving into a home that is presently occupied by frum people. They know that one can not take the mezuzot down if you know that the new occupants are Jewish. The old tenant called me to ask me to bring 9 mezuzot to replace the 9 he is taking with him. There will be a few weeks between the time they leave and I move in. I would prefer not putting up my mezuzot until I move in. The old tenant prefers that my mezuzot be up when he takes his down. Do I have time or must I do as she says?
The idea of a person not taking down the mezuzos is that the house should not be left without mezuzos.
Therefore, you don’t “have time” in the sense of them taking down theirs, and you putting yours up when you move in.
Having said this, you don’t have an obligation to put up your mezuzos before you move in, and it is officially their responsibility to ensure that the mezuzos remain up in the space after they move out, before your rental period commences.
You will be doing them a favor by bringing your mezuzos, which will allow them to take theirs with them when they move out without concern. If this is difficult, an option is to put up cheap (kosher but not mehudar) mezuzos in the interim.
Again, this is not your obligation, but because it is for your benefit, it is good to reach an arrangement that all parties are satisfied with.