Why is the Professor still in a wheel chair?

Days of Future Past reveals, albeit somewhat subtly and with more inference than direct exposition, the reason for Xavier's inability to walk...

His lower paralysis has manifested as a result of the psychological trauma of the gunshot, but his psyche has embraced it implicitly as part of his mutancy. When Xavier takes Mutant supressing serum, he regains the ability to walk: meaning his paraplegia is directly linked with his powers. In order to use his powers, he must lose the ability to walk.

Personally, I think this was supposed to be conveyed as part of his secondary-stage mutation: which was perhaps too much of a concept to fit into an already crowded narrative.

The real unanswered question, is why does Xavier still possess the likeness of Patrick Stewart, when X-Men: The Last Stand shows that body being atomized and his consciousness transferred into another... A question I intend to ask, myself!

When he dies in X-Men 3 and transfers his consciousness into his brain-dead twin brother who was mentioned earlier in X-Men 3, he is still in the wheelchair because its has become a part of his mutation. His brain/consciousness requires that he can't use his legs even though his brother's legs are fine - his brain still went through the trauma of paralysis so he won't be able to use his brother's legs.

I actually have an interesting theory about this.

There is no implicit link between the dark future timeline in Days of Future Past and the original trilogy and Wolverine spin offs other than the characters and the actors who portray them. People are assuming that the events of the dark future timeline shown in Days of Future Past are part of the same universe as the original trilogy and the Wolverine spin offs, despite a massive amount of evidence in the form of plot inconsistencies and continuity issues that this is not true.

I would go so far as to state that First Class and Days Of Future Past are a part of one X Men Universe, and that the original trilogy and the Wolverine spin offs are part of a separate, now redundant universe. This would explain why Mystique and Professor X never acknowledged one another in the original trilogy, or that Bolivar Trask is a large, alive black man in X-Men: The Last Stand but a small, dead white man in the dark timeline of Days of Future Past, which people are assuming is the sequel to X-Men: The Last Stand. It explains away how Magneto would have ever escaped from prison after seemingly assassinating JFK to go on to be the bad guy of the original trilogy without Wolverine having gone into the past to set him free in the first place, as in the original trilogy we can just assume that was never implicated in the assassination of JFK.

It is now easier to think of the original trilogy and the Wolverine spin offs as another, entirely separate universe that has no canonical impact on the new universe that was created with First Class and continued with Days of Future Past.

As brain dead from a baby, his brother never learned to walk so Xavier would have to build up his muscles in his new body to be able to walk again. That's probably a future movie when he can finally walk again.

That's crazy. So if he had cut himself while shaving that morning, his twin brother would mysteriously also get a cut just because Charles transferred his mind into him?

And if the paralysis can't possibly be a part of his mutation. His mutation was bring able to use a certain section of his brain for higher functions like psionic attacks etc. He had injured his spine while battling Magneto and that's how he got paralysed waist down. And since the spine and brain are connected, any attempt to cure his paralysis would mean losing touch with the part of his mind that gave him his powers and vice versa.

But in the new body of his brother, no such injury exists. The neurons are not damaged and all is good. So if Charles can't seem to walk, it is because he didn't try. It's been so long that he has forgotten that he can walk...maybe lost his will force.

It's just like a patient who recovers after a fracture to the leg. Charles needs therapy and care.