Hi Iain,
Don't know if you are still pursuing Nethersole House, but Google has brought me here whilst researching the Nethersole family in Kent.
They had a property called Nethersole House at Womensworld/Barham which you have already mentioned:
https://www.british-history.ac.uk/survey-kent/vol9/pp262-266There was also a branch of the family living at Charing who probably connect, although I have yet to find the link. I have a William Nethersole, gent. who was born circa 1679 and was buried at Charing 19/2/1731. He married in 1694 at Charing, Philadelphia Dering (daughter of Christopher Dering of Wickins, descended from the Surrenden family). They had:
1/ John Nethersole, b.c.1696 or c.1716 and died 1749 and buried at Charing
2/ Elizabeth Nethersole, baptised 1705 at Charing
3/ Edward Nethersole, baptised 1707 at Charing
4/ George Nethersole, baptised 1711 at Charing
5/ Heneage Nethersole, baptised 1716 at Charing and buried 1716 at Charing
6/ Mary Nethersole (uncertain is she was actually their daughter, or someone else's) who married 1735 at Charing to John Burges
Now next I have to jump to 1722 when William Nethersole married Frances Peirce at St. Margaret's, Canterbury. William was supposed to be of Charing and was a Lieutenant in the Marines. They had:
1/ Frances (d.1772) who married Rev. Andrew Portal, MA, Vicar of St. Helen's, Abingdon
2/ Elizabeth (bapt.1723 at Faversham, d.1758) who married 1748 at St. George's Chapel, Mayfair, Abraham Portal (brother of Rev. Andrew, above), who was of Ludgate Hill, London and was a goldsmith and author. After her death he remarried.
3/ William Nethersole, gent. (bapt.1730 at Faversham) - he was a wax & tallow chandler in The Strand in 1758. He married Mary Magdalen Portal (sister of Rev. Andrew and of Abraham Portal, as above). They had 4 sons and 3 daughters and may well have Nethersole descendants still living today.
4/ Thomas Nethersole (bapt.1725 at Faversham, died 1787) - he was a grocer in Soho
5/ Richard Nethersole (d.c.1774) - he was also a grocer, probably in business with his brother, Thomas. He married 1768 Sarah Hart and they had a son and 3 daughters
6/ Susanna who married 1767 John Edkins of Hampstead and late of St. Giles-in-the-Fields, grocer and had a son and daughter
I can add a little to the genealogy here, but I hope I am showing firstly that there were at least two houses in Kent called Nethersole House and secondly that a family of that name were certainly responsible for one of them (at Womenswold), and quite probably for the second (at Charing) and were quite likely related to one another.
Anyway I hope the above proves of interest and if anyone else out there is looking at the name of Nethersole then I would encourage them to post here.
Many thanks,
Caspar