Brody Peart was still in full-time education and so wasn't contracted to Basingstoke Town, and was therefore free to sign with whoever once he left school. Simmonds, I'm absolutely certain, will be on an academy contract at Albion (for this, his U-16 schoolboy season). I'm sure Albion will have...
As I said, a player on a scholarship deal can't just leave and go to another club (the scholarship is a contract and the player is registered to play for that club - the FA wouldn't allow his registration to be transferred without the original club's consent). However, once the scholarship ends...
Boys in the academy (U-9 through U-16) can sign academy contracts with clubs - I think I'm right in saying that for the earlier years it is a maximum of one year but can be two years for later years.
And yes, a player who has signed a scholarship can't just up and leave (without due cause)...