I know that new blocks are relayed one block per 10 minutes but that time is not exact but average so: Who decides that the new block is ready to hash?
A new block is ready to hash as soon as the last block has been solved. Miners immediately start to try solving the next block. As soon as one of them succeeds, that block is published and all the miners move to work on the following one.
The difficulty is adjusted so that a solution is found by somebody, out of all the active miners, on average about once every 10 minutes.
Each miner starts working on a new block, as soon as they hear about the latest one being found.
To that end, miners create block candidates and apply the hashing function. As the outputs are determininistic, yet unpredictable, this is effectively a random process where each attempt has a minuscule chance of surpassing the target difficulty.
Whenever one miner manages to find a block candidate that is below the target, they have successfully found a new block. This could happen seconds after the previous block was found, or much later. The difficulty is kept at a level such that in average it is expected to take 10 minutes.
After a block is found, the miner sends it to his peers which redistribute it. Every recipient can simply check that it fulfills the criteria, and then will start working on a successor.
In addition to the other explanations, it's important to understand that verification of a block takes significant amount of time, and it's consequences.
After a new block is found by a miner, this block is propagated to the rest of the network, including other miners. When other miners receive this news, they have to make a decision.
Should they wait to verify the block first before starting on the next block or start immediately on the next block, and assume that the previous block was legit. The benefit of making the assumption is that there won't be any wasted time spent mining blocks that will stale (orphan), which increases their profit significantly (on the order of ~10%).
As a result, many large mining pools skip this verification process and immediately starts work on the next block.