@Donna Clark - including the couple of photos that @Vic Patterson posted above of the Old Hall there are just a few of the building and some info = comments by local Bedlington Facebook group, Bygone Bedlington, members - but I haven't seen any official documents on the Old Hall.
This is a 'Then & Now' (Google street view 2020). When the Old Hall was demolished, late 1950's, the new building, Council Offices, included a section, left side, as you look at the building, to 'resemble' the Peel Tower that was part of the Old Hall.
This comment by Brian jenkinson - Bygone Bedlington Facebook Group :-
Brian Jenkinson
The "square bit" on the left started life as a pele tower which predates the hall.
The current square structure is "a nod" to its history but is contemporary with the rest of the building (opened in 1961) and not medieval!
Some more photos, but no additional info from the members who posted the photos of the old building :-
These next photos are through the archway :-
This info is from the book - The People's History, Memories of Bedlingtonshire and Beyond - by a local man, Evan Martin and t info was with the photo that Vic posted above :-