Not exaggerated, unfortunately, at least for the well-regarded schools in major cities — some genuinely do fill up months in advance, especially for the more sought-after year groups. The safe pattern is: apply the moment you have a confirmed move, don't wait for your own visa/housing to be finalised first, since school admissions often move on their own separate timeline entirely.
The Family & Education section of your destination's guide usually names the well-known schools and roughly when their application windows open — worth checking before you assume you have more time than you do.