The Blairites are Tories - that's who the BBC supports.
Seemingly.
Compared to the Corbyn supporters, the Blairites look like Tories, but there are big differences. I don't believe the Tories would ever have presided over uncontrolled immigration for example. But you're right, the BBC do support the Blairites and there's a lot of hand-wringing going on in the Beeb that their preferred choices will probably be kept out of power beyond 2020