Hi
As you said, Robert's wife Mary died in 1799 (19 October)
Burial at Lanchester 21 October 1799
Mary Lumley, age 58
Wife of Robert Lumley, Carpenter
https://www.freereg.org.uk/search_records/6833bc59865574fb7a3e442eThen Robert Lumley married Sarah Cowans in 1800 at Jarrow.
From the abstract of the licence we know that both Robert and Sarah were widowed.
Which in a way proves it is the same Sarah who gave birth to Christopher, because there is a marriage at Witton Gilbert, 4 January 1783
John Cowans
+
Sarah Errington
https://www.freereg.org.uk/search_records/6833f10d865574fb7a521c48Burial at Witton Gilbert, 13 February 1798
John Cowans, age 49
https://www.freereg.org.uk/search_records/6833f35e865574fb7a53177eIn his will proved in 1806 (which I should have read properly), Robert does name his eldest son George Lumley of Spring Gardens, London.
And if anything should happen to Sarah then George becomes guardian to Christopher.
Sarah Cowans, daughter of Robert's wife by her first husband is also mentioned in Robert's will.
So that will really confirms everything. I do think Robert Lumley junior also went to London, and remember that in his will proved in 1837 he named his brother George, living in Charing Cross.
John