How easy is it to change courses within the mathematics department

The stats department is a separate department at UCL so would entirely depend on them. Hard to say in that case.

I would suggest requesting they consider you for another course now if possible. I think there's an option to do that on Portico.

the courses are both in the maths department, so the change is all internal to one department

Joint honours degrees at UCL will have one "home" department for administrative purposes, but are co-convened by both participating departments and so admissions involves both departments normally.

In any event my point stands - it's better to request a change earlier rather than later as your chances of being successful dwindle over time as more spaces are filled on the other programme.