Actually spring isn't necessarily the most clever time to relay/reseed lawn, because it is often rather dry, though a damp spring is ideal. But at least it won't require watering as often as in summer. But if you are willing to water it regularly, then summer can be better because it will establish faster in the warmer weather. If you are less willing to water it, then autumn can be the best time because it is wetter than spring. If reseeding, by November it is likely too cold to get germination, need to wait to April.

Just laying turves down without preparing the ground is a waste of time, the turves die unless their roots can penetrate into the lower ground, which requires it to be nicely soft tilth. Plenty of sand in the soil is good for lawns to make the soil light, and because grasses actually needs sand as a nutrient. Once you've prepared the ground nicely, you may as well reseed instead of turves, much cheaper.