From purchase to moving in, 9 months is realistic, but due to the current workload of construction companies, it is not possible everywhere. It can be faster, but it becomes stressful for everyone involved. There are also differences in construction types and whether it is an architect, developer, or general contractor. A building permit or building notification also makes a time difference.
The 3-month construction period is correct, but no one tells you when they start. For some companies, the construction time is extended again in the finishing phase. "Oh, you want underfloor heating? Then it takes 3 weeks longer..."
If you plan including planning and buffer for 9-12 months, it will usually fit.