Answer: The Beraisa (Maseches Mezuza 2:11) teaches that one is obligated to affix a mezuza on a new house as soon as it is completed. There is much discussion, however, as to whether this refers to owning the house, furnishing it or actually living in it (See Shevet Halevi 6:161).
R’ Moshe Sternbuch (Teshuvos Vehanhagos 2:544; 4:440) writes that one is only obligated to affix one’s mezuzos when the house is fit to be lived in. So long as the house is unfurnished, it isn’t considered to be habitable and so exempt from mezuzos. He quotes R’ Akiva Eiger (Teshuvos R’ Akiva Eiger 141:9) who questions whether one who leaves his house for a few weeks, leaving it uninhabited, should recite the beracha over one’s mezuzos upon their return. R’ Yehoshua Leib Diskin challenges this as so long as the house has remained furnished, it is habitable and so requires mezuzos. Thus, R’ Sternbuch writes that one is obligated to affix mezuzos as soon as one furnishes one’s house with beds, chairs and tables.
The Mishna Berura (19:4) writes that one should affix one’s mezuzos just before they move in comparing it to the beracha that one says before donning their tallis. Likewise, the Aruch Hashulchan (OC 19:2; YD 289:3) writes that one can say the beracha even before moving in as they intend on moving in.
In conclusion, one who buys a new house should not affix their mezuzos until they have furnished it. While most people wait until they move in to affix their mezuzos, it is appropriate to do so from when it is furnished.