In X-Men 3: The Last Stand, which was explicitly in continuity in the timeline from which Wolverine was sent back to the past, Jean destroys Xavier's body.
The answers to How can Professor Xavier be alive? and Why did Professor X smile before being killed by the Phoenix? explain that, as shown in the post-credits scene for The Last Stand, Xavier escapes to a comatose body which is actually his identical twin (hence still played by Patrick Stewart).
But surely this twin would never have received the gunshot wound to the spine which causes his paralysis, so why is he still in a wheelchair?