The BBC have been trying for years to make interactive Doctor Who experiences work, to entirely mixed results. Even when there has been critical acclaim, the visitor numbers just aren't there.
Since "new Who" you've had two permanent exhibitions in Cardiff, lots of travelling ones, and "Doctor Who: Time Fracture". The latter was an immersive theatrical adventure which lasted two years and lost a lot of money.
Doctor Who means different things to everybody, it's part of the reason the show has been successful for so long, but it's hard to nail down. The ability to go anywhere in space and time, to have your characters refresh but remain the same, are great narrative devices for a long running show. They're not great for basing a sister attraction on.
The only permanent and familiar set is the TARDIS, but adventures take place outside it and the look refreshes every few years. Everyone has their own Doctor, and the churn rate of them can be incredibly high. Today's Doctor won't necessarily speak to fans of Ecclestone, Tennant or Pertwee. The cost effective way to do it, and not have it screen based, is to run the attraction WITHOUT the Doctor, but make occasional reference, but who wants that?
As for the cost of the IP,
@Matt.GC, if Merlin can afford Minecraft (and an upcoming Jumanji), they can certainly afford the IP rights for Who.
Once again though, I don't think that third party IP belongs at Alton Towers. The park is strong enough, in many places, to come up with its own.
I'm not particularly convinced that having a David Walliams themed is entirely appropriate for a children's area of a park either; but this isn't the Controversy section of his Wikipedia page (Hide the Sausage, really?!).