Question

What is cause of instantaneous current while the drift velocity of electron is negligibly small?

Answer 1

CHEMICAL REACTION IN THE CELL THAT GENERATES POTENTIAL DIFFERENCE

Answer 2

electrons do travel with a relativistic speed (speed that approaches or is a significant proportion of the speed of light) but only in vacuum or on its own orbit (as User explained) but not in an ordinary conductor. The transport of electron in vacuum, where there is negligible electrical resistivity results in a relativistic speed of electron. Such transport is called ballistic transport.