I must have seen that edition of QI, with its references to Corby, at least half a dozen times. (Oh, the joys of watching Dave!)
I also recall seeing the other programme you mention (although, like you, I can't remember what it was), which referred to Slough.
It seems that the second programme was in error because Slough has had a railway station continuously since 1840.
Corby had a railway station from 1879 to 1966. It reopened in 1987 but was closed again 3 years later. A new station is due to open there this December. From then on, the title will probably belong to Newcastle-under-Lyme.
Chris