A complete renovation CAN also definitely take significantly less than a year.
Requirement: you don’t do it yourself but hand it over to professionals. I just had water/sewage, electricity, bathroom and guest WC completely done in a house within two months, floors, new kitchen, partly windows and all surfaces (including lots of ceilings and also changes to the layout in drywall) done. I mainly took care of the demolition work myself (ceilings, wallpaper, tiles).
Of course, this is not a complete renovation and the house is not huge. Nevertheless, I consider significantly less than a year realistic with appropriate personnel.
And please don’t even think about wanting to live on the construction site. It is simply not humane. Noise and dust exposure are unimaginable.
Regarding the apartment search: I would say goodbye to the cat very quickly. That drastically increases your chances of finding something. If it is still not possible, the uncle has to bite the bullet. You can hardly sleep with your family at the train station, so he just has to deal with it.