Surprisingly, Lord Sugar appears to have a huge amount of the decision-making power himself. It was apparently him that decided to change the name of Junior Apprentice to Young Apprentice because apparently 'they're not junior to anyone'. It was also apparently him who demanded that the format change from Series 3 onwards so that he could fire more than one person (after the fiasco with the chickens in Series 2 when he wanted to fire Alexa and Syed, but could only fire one of them), and he who changed it from a job to a business opportunity.
I'm not sure how he manages all this, but I think it's possible that he has Mark Burnett on his side, who owns the entire Apprentice franchise. Maybe Mark Burnett can overrule the BBC and take it to another channel? I don't know exactly how it works, but from what he says, Sugar is apparently able to take his contract wherever he likes.
Sugar is an integral and pivotal part of the show, of course the BBC will listen to him and take his opinions into consideration regarding how the show should be. That's not the same thing as being in control and having the rights to the show, which belong exclusively to the BBC in the UK. Junior Apprentice is the same, as they're using The Apprentice brand and the concept is almost completely the same, just with younger candidates.
It was the BBC who chose Sugar in the first place, they initially wanted Phillip Green but he rejected it.
If Sugar decided to move to another channel and do a similar show there, I guess the BBC could sue him back into the stoneage for infringing their rights to the show. Unless he would change the format to such a degree that they couldn't make such claims, but then it would be a completely different show.