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Hi
I think that mostly the directories on film on FamilySearch are locked up and can't be seen at home. There are exceptions, and Pigot's 1823/4 was one of them.
They also have catalogue entries for directories that they hold in other formats.

So it's really down to what is available for free online, plus the many directories on the subscription sites (especially ancestry for London)
John

Also noticed more christening records for 1835-1841. Parents are Thomas Mole and Mary, church is the same that the prior Eglingham christenings took place. Although Mary Grey/Mole would've been about 52 at the latter point 

Father a grocer / shop keeper on the baptisms
Maybe
MOLE, JAMES       
Mother's Maiden Surname: CHRISP 
GRO Reference: 1839  M Quarter in ALNWICK UNION  Volume 25  Page 213

MOLE, ELIZABETH       
Mother's Maiden Surname: CHRISP 
GRO Reference: 1841  M Quarter in ALNWICK UNION  Volume 25  Page 236

Their children are listed as born in Eglingham (1809, 1811, 1813, 1815, 1823, 1826, 1829/1830) and Whittingham (1818, 1820)

Only the first baptism, Elizabeth's in 1809, was at Eglingham.
Other than the two at Whittingham, the rest were at Edlingham.
The marriage in 1808 was at Ellingham.
Confusing!

I don't have Pigot's 1823 London Directory, but from the 1825 edition:-

1823/4 is on FamilySearch
https://www.familysearch.org/ark:/61903/3:1:3QS7-99DV-HX4T

Directory starts on image 769
https://www.familysearch.org/ark:/61903/3:1:3QSQ-G9DV-HXJK

Portsmouth, Portsea and Gosport here
https://www.familysearch.org/ark:/61903/3:1:3QS7-89DV-HDK4

I have the death of Francisco Samuel Short, 9 June 1954 at the Fountain Inn, Liskeard in Cornwall.

Pedantic I know, but probate calendar says 5 June 1954, as does the death announcement in The Times (published on 9 June!)

Death announcement of Mary, 4 November 1848, Newcastle Guardian

In Barrack Square, on the 27th ult. aged 70, Mary, widow of Mr Joseph Walton, late Barrack Serjeant at this garrison. She was mother of 14 children, grandmother of 51, and great-grandmother of 10.

WALTON, MARY       
Age at Death (in years): 70 
GRO Reference: 1848  D Quarter in NEWCASTLE UPON TYNE UNION  Volume 25  Page 243

Digital image available for £3, but you may have it already.
Elizabeth goes on living in Barrack Square, in 1851 she may be enumerated as Watson, with her age greatly out. Age 59! School Mistress. Mary Ann with her.
I imagine the enumerator must have had trouble with the handwriting
https://www.familysearch.org/ark:/61903/1:1:SG67-MSS