Why does the hydrogen and oxygen gas mixture in a collection bulb not react as soon as it's collected?

When hydrogen and oxygen are collected together in the same vessel they don't immediately react unless the gas mixture is in contact with a surface that's above the autoignition temperature. This is the lowest temperature at which a substance will spontaneously ignite at standard pressure. The autoignition temperature of hydrogen is about 530 degrees C.

In order to start burning at a temperature below the autoignition temperature, flammable materials need a source of ignition such as a flame from a match or a spark from an electric igniter. The energy that must be provided to initiate a chemical reaction is called the activation energy. Once the activation energy is provided by the ignition source an exothermic reaction is sustained by the heat being released. 

Flammable gas mixtures are unstable even at temperatures below their autoignition temperature. They can be accidently ignited by a discharge of static electricity. It's also possible that, even at a lower temperature, a collision between molecules could occur at a speed sufficient to initiate a reaction. 

The link below gives autoignition temperatures of various gases.