A gas geyser takes about 6 to 10 seconds to heat up. The heat exchanger and the piping in the gas geyser start out at ambient temperature and take a few seconds to heat up. additionally, the geyser does not ignite instantly, this adds on about 2-5 seconds. therefore a total time of 6 to 10 seconds until the gas geyser is heating up water at the expected rate.
Fan force driven gas geysers tend to ignite slightly quicker and will use slightly more gas at the very beginning to get the heat exchanger up to temperature and then go down to normal operating temperature, just like you might turn the burner on your gas stove to its maximum setting to get your pot up to temperature, then turn the burner to a lower temperature. This is very brief and does not waste gas as energy consumption is relative. What it does do is save you water by getting the water initial water temperature up to the target temperature faster so you don’t need to run the tap as long as you would have.
Natural draft gas geysers do not have the same functionality as fan forced gas geysers, however its a very small amount of water so it is not something to get concerned over unless you constantly turn the tap on and off throughout the day.