My eldest DS moved into a toddler bed at 23 months, I bought it cheap from eBay though, I wouldn't pay a fortune for a bed you won't use for long. I liked knowing that he didn't have far to fall (and he did fall out a couple of times). He moved into a big single bed when he moved rooms to make room for the new baby (he was around 2 3/4).

Definitely stop using your cot if you don't think it is safe.