I think it depends more on the individual broody hen than time of year.
My experience is a bit off norm. During this past summer I was raising a second batch of three hatchery chicks. For some wild reason, when these chicks reached six weeks old, a five-year old GLW hen decided they needed adopting.
They were roosting by this time, and she would let them hunker under her wings on the perch. One night I could only see two chicks on the perch beside her. I looked and the third had stuffed itself under her.
One chick was a cockerel and he's been re-homed, but the hen is still sticking pretty close to the remaining two pullets, now almost four months old. Up until a week ago, she was still feeding them tidbits when she located them.
And during this entire period, this hen was actually broody. She would emit the low, brooding popping sound typical of all broody hens as they move among the flock.