Northumberland's traditional county town is Alnwick, and some might regard it as rather harsh not to allow this as a correct answer.
Northumberland County Council has had its headquarters in Morpeth since 1981. From 1889 to 1981 they were in Newcastle–upon–Tyne (which was not in the county from 1974).
If the setter wanted to avoid any ambiguity, they could have asked "In which town has Northumberland County Council had its headquarters since 1981?"
There are several other English counties where the same caveats apply.
© Macclesfield Quiz League 2018