At what age do babies grow out of a cot/bed?

Generally they can sleep in a cot until about a year old and cot beds until 5 years. I'd go for the cot bed.

Some babies outgrow a crib as early as 8-9 months because they climb out. Some contentedly stay in them until age 3-4 years.

There are two kinds of convertible cots, some turn into beds that take a standard crib mattress, and some turn into beds which take a standard adult mattress (I think a queen?) because the sides of the crib become the head and foot of the bed.

My 4 year old can still fit on a crib mattress with a reasonable amount of room between his feet and the end, but he doesn't much like it. There's not enough room to roll around.

Another thing to consider is that with a bed the size of a cot mattress there's no real room for an adult. So lying in bed reading stories probably isn't going to happen, and comforting a child who is sick or had a nightmare can be pretty uncomfortable.

Another thing to consider is that room-sharing is recommended to reduce SIDS so you might want a crib that will fit in your room. Bed-sharing is a more controversial option and both a cot or an adult mattress (twin or larger) can be used to provide extra space for bed-sharing.

I think every child is different, my niece could escape her cot by 11 months so she had a toddler bed and slept fine in there but my daughter had a toddler bed and a cot in her room and chose to sleep in her cot until a few days after her third birthday. Even when placed in her bed i would wake up to find her in the cot.

Cotbeds will be fine from birth-5/6 years depending on how tall your child is.