It would be pretty stupid of the new producer to come in and axe a load of them. This forum isn't generally reflective of the wider public where they're still a popular family and have been since they were introduced.
With that said the writing for them hasn't been great for a while so hopefully one of SOC's priorities will be to fix that. Mick in particular needs a lot of work; the Nancy storyline and his current treatment of Lee don't paint him as the sympathetic family man he was when he was when he first came into the show but rather a weak and childish individual. He needs a storyline that really tests him and hopefully when Linda is away it will be the making of his character.
The writing for them needs to stop going round in circles; they can't keep doing the 'What's wrong L?' storylines that DTC was so fond of. When they write for Johnny and Lee, they actually need to write for THEM, not for Mick and Linda who always end up as the main protaganists whatever happens with the family.
I would keep Mick, Linda, Shirley, Lee, Johnny and Babe and I would keep Lee and Whitney together. Lee gets plenty of criticism on here but I find him to be a sympathetic and believable younger male character and I feel there's a lot they can do with him. Johnny is in desperate need of proper characterisation because he simply hasn't been written as anything other than a spectator so far, but despite that he has a lot of potential, especially as he's training to be a lawyer which opens up tons of storyline opportunities within a few years time. He also needs a big relationship to get the fanboys and girls hot under the collar. People on Twitter and Tumblr are desperate to see this happen for him and while I don't think SOC should be actively chasing any type of audience, it can't hurt to have some vocal support through these types of channels. Babe could be redeemed, I think, or at least made a BIT less evil, and I'd rather she was as Annette Badland is such a brilliant actress and it would be a shame to lose her. I think people generally agree that Mick, Linda and Shirley are working and I do agree though as I said before, I think Mick's character needs better planning for.
I'd get rid of Tina when Sonia leaves, and make Sylvie recurring only like Elaine currently is. Those types of larger than life characters work in small doses. Buster is already going but I'd have got rid of him as well.
Don't get me wrong, I definitely think the Carters need work, but when you look at the state of families like the Beales, the Masoods (or just the one Masood

) and the Brannings, it seems pretty silly to me to axe a load of members of a family that actually still function as a family.