Timing
The best window is late autumn, after the leaves have finished falling and before sustained winter rain. Clearing too early often means the gutters fill again within weeks.
What winter does to a blocked gutter
Wet debris holds water, and a full gutter is heavy. Freezing weather adds to that load and can pull brackets loose or open up joints, turning a cleaning job into a repair job.
Worth checking at the same time
While the gutters are being cleared, it is a good moment to check downpipes and joints, so anything that would fail in the first heavy storm is picked up in advance.