In the 1881 census Thomas Lamb aged 29 (b c1852 Belford) is the innkeeper of the Harbour Inn, Amble. The Lamb tribe did come from Belford, so I suspect this is a son of Thomas Lamb (senior) that established the Warkworth Brewery.
Interestingly not only is he an innkeeper, but also a 'carpenter employing two men'. I think it's looking likely that he is the name behind Lambs Terrace.
Here's an advert from 1886, has him as a builder selling off flats in Church Street: