I have found the facts for the scenario where a person is doing two jobs, one physical and one deskbound. Signed off sick for the physical job but still able to do the other job.
"The appeal was upheld by the EAT who found that the dismissal was unfair. They held that an employee can be medically unfit for work under one contract, while still capable of work under another contract. In this case Ms P was not paid twice as there was no overlap in the hours of the two roles. The original decision to dismiss on the basis of fraud was therefore legally and factually misconceived."
The important bit is 'no overlap in the hours of the two roles'.
http://www.bevanbrittan.com/articles/Pages/TwojobSonesicknessabsenceonedismissalseptember2011.aspx