To achieve such type of functionality, i would recommend you to use AppWarp S2, an On-Premise server which will give you the full control to manage all the things on the server itself.
Below is the link to the game which has similar kind of functionality.
With AppWarp Cloud, you will have to manage the timer things on the client side youself.
Eg. Based on the condition (as soon as nth player has joined the game), you can start the game from the client side, and then start a timer of 60 sec and once the specified time interval has been completed, every player will send his result using sendUpdatePeers and then wait for 10 secs and again start the game for next 60 sec and so on.