If a person is going to have an allergic reaction to a flea bite, which almost all people do, it will happen fairly quickly (within a matter of minutes).
So yes, a flea bite will start to itch almost immediately after being bitten.
The flea does inject a mild anesthetic into the skin in order to numb the area so that the victim will not detect being bitten and therefore interrupt the flea’s meal, and it does take a little while for this numbness to wear off.