Is it recommended to buy a new mattress when pass on a cot from one sibling to their younger sibling?

A new mattress is not required for each sibling as long as the mattress remains firm, fits appropriately & is clean.

Red Nose has the following information.
Any previously used mattress should have been well stored, firm and clean.
If the mattress is damp, torn or mouldy or there is any doubt about its safety, including fit, it should not be used.
https://rednose.org.au/article/second-hand-mattresses