The full names of the companies varied over time, as amalgamations took place (rather like trade unions today). The following list, in contemporary spelling, is taken from Eneas Mackenzie’s Historical Account of Newcastle upon Tyne, dated 1824, which is likely to be reasonably accurate;

Bakers and Brewers
Barber Surgeons (spelt as Chirugeons in Mackenzie)
Butchers
Colliers, Paviors and Carriagemen
Coopers
Cordwainers
Feltmakers, Curriers and Armourers
Fullers and Dyers
Hoastmen (generally spelt Hostmen today)
House Carpenters
Joiners
Masons
Masters and Mariners (also called Trinity House)
Merchant Adventurers (incorporating the Drapers)
Millers
Plumbers, Pewterers and Glaziers
Ropemakers
Saddlers
Sailmakers
Scriveners
Shipwrights
Skinners and Glovers
Smiths
Tanners
Taylors (i.e. Tailors)
Upholsterers, Tinplateworkers and Stationers
Weavers

(This is a fuller list than on the Freemen’s website,  because that includes only the Companies that still exist)