The map here from 1897 clearly shows the shape of the porch on the south side of the building. The house is not easily seen from the road but the porch and upper central window in my photo taken from the public road (below) appear to be a good match. If it is the house, the other windows must have been widened.
Alec and Winnie Watson gave the following description:
The house (now called Four Winds), opposite Towne Gate, was Town Farm. That farm owned the buildings now in Tulip's Yard which were the farm buildings. Alec would go along to Town Farm with a milk can to collect the milk.
Caroline found out that Mr J W Stephenson, the recipient of the postcard, was born in 1869 in Newburn, and had 11 siblings. The youngest of these were still living with his parents, Sarah and William Stephenson, at Town Farm, Heddon-on-the-Wall, in 1901. She speculates that it his parents and younger siblings that are posed for the photograph outside Town Farm farmhouse, and she could well be right in that too. Caroline told me that she pieced this all together from the 1871, 1881, 1891, 1901 and 1911 census.
The brief message (given in full on Caroline's post) is signed 'Cissie'. Although this was often applied as a nickname to several different girl's names, it is often applied to Sarah and that would fit too, inviting her son to visit on Sunday. Caroline found out that John also had a sister called Sarah, who would have been about 27 when the postcard was sent. Perhaps she was Cissie...? The only thing we don't know is what they had for Sunday dinner.
A great feat of detective work and a very interesting find. Many thanks for getting in touch, Caroline.