Most cots convert to a cot bed or toddler bed somewhere between 18 months and 3 years, but the better signal than age is whether your toddler has started trying to climb out. Once that starts, the cot’s safety advantage (containment) is already gone, and a fall risk has replaced it.
A sudden move can also coincide badly with other big changes — a new sibling, starting nursery, moving house. Where you have the choice, many parents find it easier to make the bed transition separately from other big changes, rather than stacking them together.
If you do move early because of climbing, a simple floor-level mattress or very low toddler bed in a childproofed room reduces the consequences of a fall while your toddler adjusts to the new freedom.