PS it might help in your researches on the artist to know that Capt. Robert Alexander Lambert died 21 December 1805 - the Gentleman's magazine has it right, other sources including the peerage and the Oxford DNB entries on his sons have it wrong.
I see from the national Portrait Gallery: " In 1804, Jackson moved to London. He began studying at the Royal Academy schools and by 1807 had established himself as a portrait painter"
The possible date of the paintings therefore, if we have the subjects and the artist right, would be 1804-1805. Capt. L (as his wife calls him in her diaries) would have been 72-73 and Catherine 52-53, which seems to fit with the aspect of the subjects in your paintings.
For what it may be worth, I have swiftly searched Catherine's diaries for any mention of Jackson's early patrons Phipps, Beaumont and Frederick Howard, 5th Earl of Carlisle, on the off-chance there might have been an even earlier acquaintance with the young Jackson, but no trace of any of them (though a couple of years later Catherine was a friend in Brighton of Lady Julia Howard, wife of Earl of Suffolk)
There is no mention of Jackson in the diaries. There is a diary for 1804, but there is no diary for 1805 (I suspect because of the trauma of Capt L's sudden death Catherine or a descendant may have destroyed it; there is no diary for the year in which her son Capt Henry Lambert RN was killed, either). So - I'm speculating that the portrait of Capt L. was done in 1805, the year he died.
