Autumn is the best window here
From September to late November the Hampshire soil is still warm from summer while the air has cooled and the rain has returned. Roots take hold quickly and you may barely need to water at all.
A lawn laid in autumn also gets several months to establish before it faces its first dry spell, which is the real test of a new lawn.
Spring is a close second
March to May works well, provided you commit to the watering. Soil is warming, growth is picking up, and the lawn has time to knit before summer. The risk is a dry April, which means dragging a hose out daily.
Summer and deep winter
High summer is possible but demanding. Turf laid in July can need soaking twice a day, and one missed weekend can shrink the rolls and open the joints.
Mid winter is fine for the ground preparation and rarely a problem for laying, as long as we are not working frozen soil. Turf laid on cold, unfrozen ground simply sits and waits for spring.
The first month after laying
Water daily for the first fortnight unless it rains properly, and enough to wet the soil under the turf rather than just the blades. Lift a corner to check.
Stay off it for two to three weeks. First cut at around three weeks with the mower on its highest setting, taking only the tips, and never more than a third of the height in one go.