The only factor that affects *time* to climb is whatever rate of the climb the plane happens to be doing. Rate of climb is nothing more than speed measured vertically, instead of along the flight path as with IAS. Aircraft have an airspeed that will provide the maximum rate of climb for a given set of conditions & configuration. Change those conditions or configuration, and the speed that provides the best rate of climb will also vary. Fly at a speed faster or slower than the appropriate speed for best rate of climb under those conditions, and rate of climb will be reduced.
Because rate of climb is only concerned with vertical movement, anything that moves the plane horizontally, such as wind, has no effect on the rate of climb.
As a thought experiment, think of a helium balloon at floor level in a sealed container on a train that isn't moving.
* Release the balloon at floor level and it will climb at a particular rate.
* Start the train moving, release the balloon again and it still will climb at the same rate.
* Now imagine a wind that exactly matches the train's speed & direction. What happens in the container if you release the balloon again? Nothing different to before.
* Now imagine there's a window in the container's side. If you put your hand out the window you wouldn't feel any wind *because the train & the wind are all moving in the same direction at the same speed*. The balloon's rate of climb is unaffected.
* No matter how much of the container's structure is removed - a bit at a time until it's all gone, for example - there is still no effect on the balloon's rate of climb. No matter how fast the train goes, or the balloon within the moving airmass, the horizontal motion doesn't affect the vertical.
If you were standing at the side of the rail line watching the balloon's *angle* of climb through all of this then you would see a dramatic difference in its angle of climb. 90 deg when still, and becoming shallower & shallower as the horizontal speed increased. Whether the balloon's horizontal speed was caused by the train, or because it was within a moving airmass is irrelevent. Angle of climb is a measure of 'rise' over 'run'. The greater the 'run' for a fixed 'rise', the shallower the angle.