Do warts show up right away? after contracting the virus?

Warts: big warts come from the infection with a virus of a single cell. As the cell divides it gets bigger. So yes they start immediately but some grow faster than others. Big warts always come from small warts. Small warts come from even smaller warts etc.

No: One's eyes are not that sharp. Warts are caused by viruses which cause skin cells to grow abnormally in a lump. Usually, it is weeks or months after a new wart has been growing before a person can see the wart. In the beginning, it is just too small to be seen.